EMP grenades stun mechanoids, improvised turrets, mortars, deplete personal shields, and detonate mortar shells, but have no effect on humans and do not deal physical damage. A target that is stunned shows a mote saying 'Stunned'. After being stunned, targets may randomly become 'Adapted' and gain stun immunity. This stun immunity is not permanent, but takes a long time to wear off (usually lasts longer than the fight that caused it).
EMP grenades are very effective against mechanoid centipedes. Their slow moving speed and large size makes them an easy target. They have a very high damage potential, so even temporarily shutting down their ability to attack will give your colonists an advantage in a battle.
It is a good idea to manually aim EMP grenades instead of letting the AI do that. AI will not lead the targets and will not attack targets which can only be hit by area of effect. Repeated applications of stun while the target is already stunned will cause stun adaptation to happen earlier, while not affecting stun time significantly.
Using EMP grenades against friendlies counts as an attack even if they aren't hurt.