The center of the board is its crossroads, much like the center square in tic-tac-toe.
From the standpoint of chess, once you control it, you (usually) limit your opponents' options, including his power to attack you. That's a major benefit.
As to how to attack your opponent with that advantage, study some tactics.
One way is to attack a target with all your pieces, force your opponent to defend, then move some of your pieces to a different target, probably on the other side of the board. You can move faster than your opponent because you control the center, so you'll "hit" the second target.
A good exposition of these ideas can be found in "Chess Fundamentals" by former world champion Jose R. Capablanca. It is an "old" book written by someone born in the 19th century, but it taught me how to play chess years ago. It can also be found online under Googlebooks.