If you assuage an unpleasant feeling, you make it go away. Assuaging your hunger by eating a bag of marshmallows may cause you other unpleasant feelings.
The most common things that we assuage are fears, concerns, guilt, grief, anxiety, and anger. That makes a lot of sense — these are all things we seek relief from. The word comes from Old French assouagier, from the Latin root suavis, "sweet" — think of adding a bit of honey to something unpleasant. A word with a similar meaning is mollify.