This article lays it outMost significantly, Cheney was the architect of the Iraq war. Many powerful cabinet positions were his longtime allies, such as Rumsfeld.
Add to that the fact that he recommended himself to be VP.
So that makes him influential, but other people have been influential. Mind you, the VP job has generally been pretty quiet, at least publically.
Some was his public persona. He didn't wear the mantle of authority very well. He liked goofing around - giving reporters playful nicknames. He was often "misunderestimated" when it came to intelligence.
He also never had a powerful Chief of Staff who normally wields the president's power and agenda significantly.