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One reason for this is that from the center of the board, your pieces have quicker access to the other parts of the board. Another is that your pieces often attack a larger number of squares. What others are there?
I often find myself having control of the center, only to end up thinking "Ok, now what?". What advantages do I get, and how can I make something useful out of these?
In addition to the points you made, controlling the center means that your opponent will then have to form his or her attack on the wings. Think of not only what it does for your pieces but what it means for your opponent's.
Picture the movement pattern of a knight - if you control the center, the knight can attack all eight squares. If your opponent's knights are forced along the side of the board they're likely going to be in the first, second, seventh or eight columns (files). That means they're less effective. ("A knight on the rim is grim.")
If you have control of the center, it's usually the time to start planning an attack. Has your opponent castled? If so, start getting your pieces aligned to make an attack on the appropriate side. If you're playing more of a positional game it's time to start marching forward, taking up real estate on the board and placing your pieces and pawns onto squares that make it difficult for your opponent to move/develop his or her pieces.
Controlling the Center:
1) From the beginning of the game until the King castles, the center of the board is where the King is, and so it is the logical place to begin operations. After the King does castle, the operations tend to focus on creating an attack on the flank where the King has moved to.
2) The middle of the board allows for the most mobility for all of your pieces but the Rook. Knights have 8 squares in the middle, 4 on the edge, 2 in the corner. Bishops have 13 moves in the middle, 7 in the corner. The queen has 27 in the middle, 21 in the corner. The more squares your pieces control, the more options you have later in the game
3) Moving a center pawn forward frees three pieces (Queen, King, one of the Bishops). Moving a Bishop pawn forward only frees the Queen or the King. Moving a Knight pawn forward only frees a Bishop, and moving the Rook pawn forward only semi-frees the Rook. So in order to get the most pieces developed efficiently, moving the center pawns are important. Once they have been moved, it is critical that they remain as a shield at least until your King has been moved to safety.
Also consider that your opponent thrives to control the center, so you controlling it derails his strategy which gives you an advantage.
Had opening books, ending books, and training/practice regimens centered (pun intended) on flanking strategies rather than strategies for controlling the center, people'd push for controlling the flanks and not think about the center so much.
But the general premise holds. Chess pieces have easier movement and larger fields of influence when in the center as compared to being on the flanks. A knight for example, when sitting on a center square, directly covers 8 squares. When sitting on an edge square, it covers only 4 squares. For other pieces this is even more pronounced.
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.
I often find myself having control of the center, only to end up thinking 'Ok, now what?'- Now use your knowledge of chess theory to form a plan. See for example this question.