Here's my understanding, and this is in very broad strokes. Anyone can correct me if I'm wrong.
The FCC has a congressional grant of authority to regulate interstate telecommunication. It must exercise its power in accordance with statutory guidelines, but it has a lot of leeway in deciding just exactly what to do in a given situation. It had previously classified the internet in such a way that it fell outside of the FCC's power to regulate (I believe, but am not sure, that the FCC had said that ISPs were not common carriers). This new FCC regulation reclassifies the internet so that it does have the power to regulate it (again, I believe, but am not sure, that they've reclassified ISPs as common carriers), and then does go on to regulate the ways in which ISPs can limit access to their networks.
The process for the agencies promulgating new regulations is arcane, and I'm not going to go into it here, but once the regulation has gone through the proper process within the agency (in this case the FCC) then it doesn't go to congress to approve it, it just goes into effect automatically. Now, people could still challenge the regulation in court on the theory that it violates the FCC's statutory authority, or if congress really doesn't like what the FCC did they could override the decision with a new law, but unless and until one of those things happens, the regulation has full effect.