DIAGNOSIS This is a classic trigger or motor brush failure. The click is the electronic switch engaging, but power isn't reaching the motor.
FIX The primary suspect is worn motor brushes. Open the drill's housing (you'll need a Torx T10 or T15 bit). Locate the two carbon brushes in the motor end cap. If they're worn down to less than 1/4 inch, replace them. Craftsman sells brush kits for this model.
IF THAT DIDN'T WORK The trigger assembly itself may have failed. You'll need to test it with a multimeter for continuity. If it's dead, the entire trigger/switch module must be replaced as a unit.
HEADS UP Don't just keep pulling the trigger hoping it'll work. A failing brush can arc and damage the motor commutator, turning a $15 fix into a dead tool.