What is the point of controlling the middle

Controlling the centre serves mainly to improve the mobility of your pieces, which in turn allows you to control even more squares. The reason for this is that although you can get from one quadrant of the board to another via the flanks or back ranks, the centre offers the quickest access to any of them, and pieces stationed there can control squares in any of said quadrants (which is why knights on the rim are dim, for example.) Note that there are many ways to control the centre, but the main one encountered is pawns. Personally, I prefer to keep my pawns a little further back and instead let my pieces do the grunt work (I've won many games through a sheer pawn advantage.)
As mentioned, control or the fight for the center usually is related to piece mobility. The greater the mobility the more possibility for attack. Less mobility not only has less opportunity for attack but also less room to defend properly.
One example is the Italian Game Evans Gambit where White sacrifices a pawn for a strong center and attacking opportunities. An extreme example:
In closed centers, players can try to get an advantage on a wing or they can try to maneuver to open the center (sometimes known as a break) to their advantage. Often it's for an open line into enemy territory.
Example of wing attack:
Example of center break: