To dampen something is to stifle it, to lessen its effect, or to moisten it. Getting a cold will dampen your enthusiasm.
This word has several related meanings, most of which make something less effective or powerful. Using a towel will dampen it in the sense of getting it wet, but dampening goes beyond things that are damp (wet). Bad news will dampen your mood, bringing it down. Boxing gloves dampen the force of punches because they cushion the blows. Anything dampened has been weakened and brought down a notch.