Why does Stan Lee play random extras in Marvel films?

Stan Lee isn't casting himself in these cameos - it's a tradition which started back in 1989 with one of the original TV Hulk spin-off movies, one that has endured ever since.

From The Trial of the Incredible Hulk - 1989

He may have a producer credit on the Marvel movies, but his involvement doesn't go any deeper than just being a beloved figurehead - these cameos are homages by the filmmakers.

A comprehensive list of his cameo appearances, plus images, can be found here.

Quoting Lee himself:

"It’s almost scary ‘cause they just give me a date. They’ll say, "Stan, come over next Thursday for your cameo," but they don’t tell me what it is. So, I show up and they say, "Go to wardrobe." In wardrobe, they say, "We want to get you a white shirt and a green sweater and a black zippered jacket," and I’ll say, "Don’t bother, I’m wearing one." And then, they say, "But, you can’t wear yours!," and they look around and get me the exact same thing, but I’m not allowed to wear my own. And I still don’t know what my role will be. I go on the set and one minute before we’re gonna shoot, the director says, "Okay, Stan, this is what I want you to do. So, I don’t know what is expected of me, until I get there, but of course, I do it magnificently."

Unconfirmed reports suggest that he will be a beauty contest Judge in Iron Man 3 and a London bus driver in Thor: The Dark World.

Because he is a watcher or works for them. Likely Uatu, the watcher that's assigned to watch Earth. Confirmed by Marvel in the Guardians of the Galaxy Vol 2.

Marvel chief Kevin Feige has endorsed the popular fan theory that Stan Lee is playing the same character in all of his MCU roles. http://screenrant.com/stan-lee-mcu-cameo-theory-marvel-kevin-feige/

“Yes, we always thought it would be fun. Stan Lee clearly exists, you know, above and apart from the reality of all the films. So the notion that he could be sitting there on a cosmic pit stop during the jump gate sequence in Guardians was something very fun – James had that idea and we shot that cameo and loved it so much, you know, you see it a couple of times in the movie. It wasn’t in for a long time and we put it back in towards the end of the process where he references that time he was a Federal Express agent – we thought it would be fun to put that in there because that really says, so wait a minute, he’s this same character who’s popped up in all these films.”