its incredibly useful but requires timing and situational awareness. its uses are too many to go into in just one post but here are the first few that come to mind:

1) Silence enemies before casting a Heal on yourself to make sure your heal doesn't get interrupted. 

2) Interrupt charge up times of teleport scrolls or healing potions to keep a weak enemy in combat and get the kill.

3) Use Silence at the beginning of a group vs. group brawl to create a 3-5 second DPS gap by delaying their ability to use nukes and debuffs from the onset of the fight. your whole team will be 3-5 seconds ahead in cooldown refreshes and will likely get in one more round of ability activations than the other team. 

4) use it as your're running away or pursuing a runner. it shuts down item activations as well so it will prevent them from using a Wand of Speed or Warpstone if they have something like that. 

the list goes on. its very versatile but its uses are purely situational. 

You'll need it if you want to kill an enemy Sed, Qot or Oak since otherwise your opponent will just heal/shield and get away most times.

Once you have it to 4 seconds this is long enough to teleport to safety, so its quite handy if you're backdooring portal flags on Cataract for example.

It's not terrbily useful against some DGs, tower Rook, Reg, UB for example won't be terribly concerned at being silenced unless they're about to get ganked because their autoattack is strong enough to really hurt.

After the silence ends, most people will use some sort of ability straight away which is pretty easy to predict so you should be able to get another interrupt with pounce