Placing insulation in your bathroom's interior walls will help soundproof the room and control the moisture. It will also help keep the temperature in the room consistent with the rest of the house. A vapour barrier is required when insulating outer walls, but isn't necessary with interior walls. When you are installing insulation, make sure one side of the wall is already drywalled so that your insulation will not simply fall out the other side.
Caulk along the cracks where the wood meets the drywall on the top and bottom using a silicone caulk. Use the tip of your finger to smooth it out and seal it.
Measure from the top to the bottom of the space you will be insulating, with a tape measure. Cut that length of insulation from the roll using a utility knife.
Stuff the insulation into the space working from the top down. The insulation should be snug. Make sure to push it in along all edges. If there are any gaps left over, cut off another small piece and stuff that in as well. The entire space should be filled with insulation.
Some insulation comes in strips with a paper backing. If you use this insulation, tuck it in as you would with the regular insulation and then use a staple gun to secure the edges of the paper to the wooden studs.