Why dry steak/meat/chicken/fish before cooking on a stove?
Because any moisture left on the surface will turn into steam upon touching the hot pan.

The steam will cause a lot of oil splatters (which is not only a huge mess, but can also hurt you), and if it can't evaporate fast enough, will interfere with proper browning, especially if you are overcrowding your pan or if your meat exudes a lot of moisture.