Basically, the banning phase exists to counter the enemy team's playstyle and strategy.
If you haven't really played that many ARTS/MOBA games, it might be hard to understand, but I'll try to explain it to you as simply as I can.
Let's say that there are 5 different main strategies teams generally use in DotA 2:
1. Pushing
2. Ganking (Assassination)
3. Team Fight
4. Balanced
5. Carry (Lategame)
Most teams usually fit into one of these categories, are there are certain heroes that make each category much easier and more effective to do.
So say, for example, that Team 1 is going for a heavy tower push lineup. The team that is facing them (Team 2) may start banning heroes that are really good at pushing so that Team 1's strategy doesn't become overwhelmingly good. However, Team 1 may start banning ANTI-Pushing heroes as well, in order to make their strategy more effective.
In other words, bans are a way to shut your opponent's strategy down while making yours more effective. It would be like banning your opponent's units in an RTS game that counter your strategy, or make his predicted strategy harder to execute.
There's also an aspect of banning known as "legacy bans" or "respect bans", which basically means banning a hero (or heroes) that someone on the other team is ridiculously good with. The reason it's called a "respect ban" is because you respect a person's ability to play a certain hero so much, that you ban that hero just so they can't have them. In the Na'vi vs. IG games yesterday, IG banned Pudge all 3 games. Pudge is a hero that is barely picked in competitive play, but Na'vi knows how to use him so well that IG was afraid to let them have it.
In summary, banning primarily exists (in theory) to make the game more interesting, varied, and strategic. However, like Lancefighter said, a lot of the time bans just come down to taking certain perceived overpowered heroes out of the game. This is very boring and stagnant, and hopefully something IceFrog and Valve continue to work on as DotA evolves.
