From my personal experience as a youngster who experienced it first hand, from beginning to "end", it wasn't the consoles, so much as it was the ability to get nearly the same experience at home. The first system to have real, licensed and popular titles, that really could stand up to the arcade versions, was the Atari 400/800 home computers. It was only after getting this system, that I felt less compelled to spend the time and quarters at the arcade I had previous to it.
I still went to the arcades, but only played games which I could not experience in some very close form to those I already had at home. What changed the arcades was their focus on trying to provide an experience which users cannot achieve with a home computer or console. The games got bigger, the control schemes wackier, etc... As the experiences delivered by home hardware approached parity with those in the arcades, the arcades became less and less relevant. Movie theaters are experiencing something similar in that industry, which is why the screen sizes went up again (after going down a number of years ago, to provide more selection), and 3D has heavy focus.