They used Comcast's standing as a cable provider, and the fact that the company uses that cable to deliver the Internet as their reasoning for being able to intervene.
Secondly, by creating a defacto net neutrality law, they've gotten into the regulation of business, which is again, outside their scope of work.
While Comcast is likely to take the case to court to get the FCC's ruling overturned, they and other net providers are already working on a way around the problem. Pay per gigabyte is being tested by a number of Internet providers in various locations.
Comcast, Time Warner, and AT&T are all looking at using a per gigabyte charge to dissuade bandwidth hogs. Considering a lot of those hogs are fairly young, and may not be paying the internet bill, it's probably a good way to go about this. Much like the parents of kids who run up huge cell phone bills take away the phone, I can see a lot of parents suddenly being interested in what the kids are doing on their computers.
Most cable providers have always had one speed for residential customers, while DSL has usually had tiered speed levels, with corresponding higher cost for higher speed. Comcast, Cox and Time Warner are all looking at that as an option, also. One way to keep the kids from downloading movies is to slow the connection enough that the movie never finishes downloading.
Other media companies, of course, don't like the idea of a metered internet; it definitely cuts into their delivery options. Netflix for example allows you to rent and watch movies online. But if 1 movie takes up 25% of your montly bandwidth allotment, they fear folks will back off renting from them.
Of course, those media companies aren't the one's paying for the infrastructure to carry all of those movies, XBox live games, and streaming audio and video. They just want it unlimited so they can expand their businesses, and don't like the idea of their customers paying for it.