Van Persie has announced he won't be signing a new deal at the Gunners.
"I can understand why Van Persie has decided not to sign a new contract," Campbell told the Daily Mirror.
"When you go back to your club for pre-season, and you look around the changing room, you want to see top players around you and think, 'Yeah, we're going to do something this season. This is where it's happening'.
"That's what van Persie is looking for. And maybe he's had a look and thought, 'Is this it?'
"Robin has seen the good times, and been part of the good times, and he wants more of it. Basically, his statement is shorthand for saying his patience has run out, and that he can't wait for ever if he's going to win major trophies in his career.
"From the outside, it looks as though he wants to go to a club who will start the season expecting to win things, not just to be in contention with a lot of ifs, buts, maybes and strings attached."