The extremely short summary is that the Union-level governmental structures weakened and were ultimately taken over by the Soviet Socialist Republic-level governments, which in turn became independent countries.

I've answered a bunch of different questions on aspects of this: how Russia took over the army and police forces is here, how it took over Communist Party assets is here and how it took over the Soviet seat at the United Nations is here.

Let me know if there are specific questions and I'd be happy to follow up.