Our hands can bend, they can stretch, they can wiggle, and they can wave! To be able to do that, the skin that covers our hands needs to be able to adjust to the changing shapes. The lines are just folds of skin that allow the hand to bend. Observe that all the lines appear at locations where our skin folds over. They are technically called 'flexion creases', and many of them are on our hands from before we are born. When our hands bend, the creases help fold the skin. A baby will have fewer, softer, lines, since it has not moved its hand as much.
Feet are very similar in structure to hands but do not bend as much. The lines on the bottom of our feet are not as severe, because sole of feet does not have so much flexibility as hand.