Beastmaster is a versatile hero who can fulfil a number of roles for his team - mainly pusher and initiator.
+ Skills are effective against magic immunity
+ Strong pusher
+ Provides extra vision for the team
+ Powerful stun
+ Can stack/farm ancients from early on
+ Good base damage and armour
+ Strong passive aura
- Mana dependent
- Slightly weak overall stat gain
- Can struggle to find a defined role/laning position
- Lacks the damage to be a true carry
- Somewhat farm and level dependent
An interesting AOE nuke, Wild Axes have some good additional utility. Every unit in their path take damage, and can be hit once PER AXE. So, it does up to 180 damage at level 1, but only if the target is hit by BOTH the axes. Due to the path the axes use, this means only targets at close range or the apex of the throw will take full damage. It takes some practice to get used to the path of the axes, and a little skill to land them consistently against moving targets.
Wild Axes also cut down any trees in their path - useful for cleaning lines of sight on the offlane, and also for prevent heroes from juking or hiding in the woods. They can also let you escape from Sprout easily.
The final interesting property of the axes is that they deal an unusual form of damage known as "Composite". This damage is reduced by both armour and magic resistance, but goes straight through both magic immunity and damage block.
This means that it will still damage heroes using Black King Bar or abilities like Rage. It also means that the axes can damage Ancients - allowing you to stack and farm them efficiently early in the game.
There is no increase in mana cost from levelling axes, the damage simply scales upwards with each point.
Call of the Wild (Boar)
It's probably fortunate they didn't name this "Call of Nature", Call of the Wild is actually two separate summoning spells that you level together. One summons a Hawk for extra vision, and the other a Boar with a powerful slow.
After the recent 6.81 changes, you can now have up to two of each of these for brief periods - summoning one will NOT destroy any existing animals. The ability is very cheap at 15 mana, but you do have to be careful not to feed the opposition with them, as they a reasonable bounty attached to them.
Using the summons properly requires some micromanagement - generally you'll want to assign hotkeys to the hawk, boar and yourself (assigning a key to yourself and the boar can also be useful).
The hawk is essentially a mobile ward - it provides flying vision over an area below it, especially useful as it can spot enemies in terrain that is normally difficult to ward. A good Beastmaster should be providing constant additional vision for his team once he has the mana to do so. The only other sneaky use I know is that your team can Roshan early by teaming up with Enigma and using the hawk to provide a source of Eidolons.
The boar is more interesting, and is available from level 2 of his ability. His attack is weak against heroes, mainly providing a very respectable 20/35% movement and attack slow on them. This is massively helpful for both escaping and ganking, and you should use the boar in addition to your own attacks here.
The boar also provides the usual minion duties - scouting, pushing and stacking. If you're not using him for anything else, stacking ancients for yourself or your carry is always a useful job for him.
Inner Beast is your passive skill, providing a respectable attack speed bonus to any allies within a fair radius. This goes quite a long way in helping with your relatively poor agility gain, but is just generally very useful for your team.
Carries will appreciate it, especially if they have poor agility gain ( Tiny, Doombringer, Outworld Devourer etc).
It's useful for pushing, providing additional attacks for your minions and nearby creeps. Obviously we have to be careful choosing when to level it - too early and it will push your lane out.
Primal Roar is a really powerful and reliable single target stun, however it can also damage and slow nearby enemy units.
The primary target will be stunned for up to 4 seconds, even through magic immunity - the only thing that can block it is a Linken's Sphere. The movement/attack slow will also go through magic immunity, and is extremely potent at 50%.
Any enemy targets hit will also take damage, but this will NOT go through magic immunity like the other effects.
This stun is one of the main reasons for picking Beastmaster, and is useful for both ganking and teamfights. Aghanim's Scepter reduces the cooldown on it even further, and also increases the cast range.
The builds above are reasonably standard, with Wild Axes usually the first skill to be maxed, with early points in Call of the Wild for vision and utility. I haven't put any builds up for it, but sometimes you will skill Inner Beast as soon as level 4 if you're in an early push lineup.
The main thing to bear in mind when skilling Beastmaster for me is how much mana you have available. Wild Axes is expensive compared to your mana pool, and you're not going to be able to use it much before you get a Soul Ring. Call of the Wild is very cheap to run, and level 2 gives you access to the Boar, with it's slow and stacking potential.
The slow is very useful - it provides a much better chance to kill most offlaners than the bit of extra damage from level 2 Wild Axes. You can also use it to stack the jungle, or stack ancients cheaply if you can't afford to Wild Axes them every minute. If you do have mana, then max Wild Axes ASAP in most cases.
Inner Beast can be levelled at 4, as noted, but you'll generally leave it until later. The rule of thumb is that you want at least 1 point in it when your team is going to seriously start pushing towers - so you may well get it before 10 depending on your lane and team composition.
Primal Roar should be levelled at every opportunity, the mana cost increase is really minimal, and everything else is better.
This is where Beastmaster really causes confusion for most players. He's versatile and does a number of different jobs...but what exactly is he?
The answer isn't straightforward - he really is jack of all trades, master of none. He can push, initiate, disable, semi-carry, provide utility and aura stacking. There are lots of heroes who can do all of these individual tasks better - but very few who can turn their hand to so many.
A better answer is that he's usually a #3 role (#4 for support) - he tends to play mid or offlane, and tailors his items to whatever the team needs most. This is normally a push, utility/aura or initiator build, with a slice of semi-carry on the side. You tend to pick up whatever responsibilities your team needs filling the most.
For example, if your team didn't have a proper initiator, you'd probably pick up a Blink Dagger and Aghanim's Scepter so that you can provide that. If your team has an AFK jungler and only 1 support, you may well fill in with a Mekansm if the core heroes aren't very interested in one.
Whatever role you end up in, using your Call of the Wild minions as scouts/wards for your team is essential.
Initiator / Ganker:
You should be mid or offlane for this build, to give you reasonably fast access to Primal Roar. You want to rush a Blink Dagger after relatively few other items, and then progress into Aghanim's Scepter afterwards. Some games you may be able to get away with a Force Staff, which provides some nice stats and an escape, but generally a Blink Dagger is preferred for perfect placement of Primal Roar for maximum effect.
+ Suspect enemy team will roam heavily
+ Need a hero with strong disable/pushing skills
+ Going for an aura stacking team
+ Need magic immunity busting stun
- Team lacks other pushers
- Nowhere for him to lane effectively
- Want a specialist for a particular role
- Heavily countered by opposition lineup
Beastmaster is most commonly played as an offlaner. He's tanky, has an AOE nuke and is not super-farm dependent, so can do a job here. What he really lacks is an effective escape - the closest you get is the slow from your Boar, and the early warning from your Hawk.
Some people do like to go for a Tango/ Boots of Speed build for offlane - this is quite good if the opposition only have stuns instead of slows, as you can generally outrun them early on, and position yourself better. However, against strong slows this is a bad idea.
So...he's not an especially brilliant offlaner...melee, limited mana pool...etc. Why play him there? Part of the answer is that the offlane provides reasonably close access to the Ancients camp, especially if you're playing as Dire. Beastmaster can use the composite damage from Wild Axes to boost his farm considerably by stacking them up. If this goes well, you can actually emerge from the offlane somewhere near the average gold/xp for a mid player.
First things first - how do you stack ancients? Simple, you just need to attack them and draw them out of their camp at xx:50 - xx:53 in the minute, extra ancients will spawn in the empty space at the minute mark. You do this yourself with a melee attack, use Wild Axes, or the attack from your Boar.
Keep stacking them whenever possible while you build up the money for a Soul Ring, and then use repeated Wild Axes to farm them safely.
How you do your starting items, and how much time you spend in the offlane all vary, as does your build order. Until you can actually start killing the ancients you'll not be getting farm/levels from them, so you need to hang around the offlane when not stacking, picking up what you can. The big farm/xp boost is all backloaded into when you can actually kill those ancients, and higher levels of Wild Axes make it quicker and easier to farm.
Tailor your starting items and build to what you expect to face in the offlane - if you're maybe 1v1 with a relatively weak solo then stack the ancients when you remember, but concentrate on winning the lane. If you're up against a tough trilane, focus far more of your time getting that stack nice and big.
Generally going 2 early levels in Call of the Wild allows you to stack cheaply (15 mana) using the boar, and also saves you having to leave the lane if you're playing as the radiant. However, this means the ancients will take longer to farm as you don't soften up with Wild Axes as you go.
Your starting equipment also has to be appropriate - you'll need Tangos, Healing Salve and a Stout Shield if competing for hits against most lineups...but if you're completely outgunned you may just decide to go for Claritys and a headstart on your Soul Ring.
Be aware that some teams will recognise you as a stacker, so be prepared that they might not be available, or that you may need a Sentry Ward from one of your supports early on (if you're lucky enough to have proper supports who'll buy them).
There are just so many items that can be useful on Beastmaster that I'm not even going to attempt to cover them all. The builds above cover the vast majority that I would consider the most useful, and to try to give you a flavour of the kinds of builds you'll generally want on him.
In this section I'm just going to provide some justification for the starting/core items, and then talk about the more interesting selections.
Heroes who attack with massive damage, but slow attack speed, will benefit greatly from the boost provided by Inner Beast.
Beastmaster benefits greatly from mana batteries on his team, allowing him to fully utilitise his early game abilities.
Other heroes with reasonably strong minions also stack up nicely with Beastmaster, providing additional units which can push strongly with his aura.
This guy is really bad news for you, especially early on. A strong slow, combined with extremely potent harassment and the ability to wreck your ancient stacks with Plague Ward.
Again, strong ranged harassment makes you a spectator when laning against them, and they're also pretty effective at slowing down your push.
When you go to all the trouble of disabling an enemy for 4 seconds, the last thing you want is them suddenly feeling better after a much shorter time.
EDIT - Thanks Aba43.