They don't necessarily stop people from cheating. But if the administrator of the server is active, they can ban cheaters from the server and never allow them to join again. See if you can kick people but they can just rejoin over and over, you gain nothing. But if you have an administrator who has the ability to ban them from a server for life, then you have something to build on. There are not a lot of cheaters out there. So if you actively ban them, pretty soon you have locked the cheating community out of your server, and people will notice that your server is a good one, and they can join it without worry of cheaters being on board for the most part. Not 100%, but your chances are really high of having an enjoyable gaming experience.
The second part is, dedicated servers don't have to run the graphics portion of the game, so they can respond faster, and host more people. 9 vs 9 is actually a very small match in the scheme of online gaming these days. On TF2 with dedicates servers it is not uncommon to see 16 vs 16 or so. I have even seen 64 man maps running flawless.