You're able to ban champs in Legaue of Legends, because DotA (to which League is the spiritual successor) allowed you to.
But I doubt that's the answer you were hoping for.
In so-called "Draft Pick", you cannot have the same champion on each team. They are exclusive - if Team A picks Shen, Team B can not pick Shen, and vice versa. Bans work similarly, in that, once banned, neither team can play the champion.
Banning is not necessarily due to the strength of the champion; you also have to consider how well that champion is actually played. Put another way, a strong champion played by a poor player is not strong, and an average champion played by a superb player might prove unstoppable. Banning invariably represents a strategic decision on behalf of the team captain, especially in the high-level tournaments where the pro players are relatively well-known.
For instance, George "HotshotGG" Georgallidis is well-known for playing Nidalee. Teams playing against Counter Logic Gaming may wish to ban Nidalee, not because she's overpowered, but because she's overpowered when played by HotshotGG.
The goal is not to prevent Nidalee from being used in the game - the goal is to prevent HotshotGG from playing Nidalee. Selecting the champion you don't want the enemy to use is one way to do this, but if no one on your team plays the champ as well as your opponent (such that picking the champ for your team would be a liability), the other way to prevent that champion from being used is via banning.
Tactically banning (or beating them to pick) contested champions can force teams to shake up their roster, by either playing inferior compositions, or less-familiar champions, both of which grant the banning team an advantage.