Any spell currently being cast will be prevented from completing when the target is silenced. The spellcasting of non-player targets is also fully interrupted, allowing targets that are immune to silences, such as bosses, to still be interrupted by this ability.
In PvE, there are many caster-type mobs that can't be pulled like normal mobs. Pulling with Silence will cause them to come to you to engage in melee, since their ranged attack (magic) is disabled. It can also stop a mob's healing, which is sometimes very useful.
Patch 1.6.0 (2005-07-12): Several creature spells that silenced players are now properly flagged as silence effects, which will allow talents such as Unbreakable Will to provide a higher chance to resist the effects.