The Swarm is a light-damage teamfight and farming ability that summons a swarm of 12 beetles that latch to targets, dealing damage and reducing armor every second. These beetles take the aggro of creeps almost immediately, making it useful for pushing lanes. The armor reduction is what makes this skill shine in teamfights, and a well placed swarm can cause effective chaos in the ranks of enemy heroes. A poorly organized enemy team will divide up between killing the beetles, wasting time, or ignoring them, allowing armor to be reduced and damage to be dealt.
It's good to note that these beetles last about 20 seconds and deal 30 damage each second they attack. This can render up to about 600 damage, maybe more if they reduce the victims armor to a negative level. Unfortunately, this skill is mana intensive (for Weaver anyways) so it's usually a good idea to leave this skill until late game and leave Weaver's mana pool intact for using Shukuchi and Time Lapse.
Shukuchi is Weaver's main skill, the one that makes him a great carry and justifies his squishiness. It allows him to go invisible for up to 4 seconds and increases his move speed to the maximum 522. He also deals damage to units he passes through (150 at its highest level).
This is a great skill early game, and allows Weaver to effectively harass enemy heroes in his lane while staying relatively safe. It also serves as his primary escape skill, making it ridiculously easy to counter ganks and get away from losing fights. At higher levels, this can be spammed for the increased move speed, allowing you to move around the map freely. In teamfights, Weaver can use this skill to get in close to his target, deal damage, then escape before they can retaliate, effectively turning him into the perfect hit-and-run hero.
It is essential that you max this skill first, as the decreased cooldown will be a complete lifesaver during teamfights, escapes, and ganks.
Geminate Attack is what gives Weaver the potential for ridiculous DPS. It allows Weaver to attack twice in one attack. This ability actually dispatches another projectile, but as long as the first one hits the target, the second one will as well. Generally speaking, this allows you to deal double the damage of a normal attack in the same amount of time that it takes to perform a single attack.
Use this skill to harass enemies early game. The damage dealt will be unbearable to squishy heroes, and will serve as a good counter to higher health heroes. This is also a great tool for last hitting creeps, but remember that the double damage is not instant, but another separate projectile, so timing is everything.
It's a good idea to max this skill out second, but consider maxing it first over Shukuchi if you need more last hits or you are getting destroyed in team fights.
Time Lapse is Weaver's secondary escape skill, but it is probably his most powerful. It allows him to revert back to where he was 5 seconds ago. His mana, health, and cooldowns all revert back to where they were at the 5 second mark. This is what makes Weaver such a powerful and diverse hero. He can take a huge amount of damage, make the enemy team waste all of their spells on him, and then simply revert back to full health and mana. This allows him to do anything from tank, to dive towers early game for kills, to juking disables, clearing his debuffs, etc.
In combination with Linkens Sphere and Shukuchi, this makes Weaver almost invincible in teamfights so that he can keep hitting enemies with his insane DPS.