There isn't an in-universe explanation for these types of things.
It also happened when Don Cheadle replaced Terrence Howard as Rhodey from Iron Man to Iron Man 2. In that instance, it's even more clear that the characters are a part of the same continuity (while there can be considerable leeway given to the first HULK movie).
We see that it's being portrayed by a different person, but in the fictional reality nothing has changed.
We're clued in to that at the beginning of Iron Man 2, in which Don Cheadle first enters as Rhodey.
Stark: Hey, buddy. I didn't expect to see you here.
Rhodes: Look, it's me, I'm here. Deal with it. Let's move on.
Stark: I just...
Rhodes: -Drop it.
Stark: -All right, I'll drop it.
This scene is explained in an interview with Cheadle by Collider to have been a reference to the casting change:
Interviewer: I’m curious about the first scenes you guys shot were the courtroom scene with the senator with Gary Shandling and obviously you’re coming into a role that originated with another actor. How daunting is it or how was it when you first walked on set with that many people and try to still figure out how am I going to do this?
Don: Well, a lot of that was figured out before we began shooting. And really there were no marching orders of, watch the first movie and make sure that you come in here and you’re paying off what Terrance did in the first movie. It was really you’re in character; you’re going to do your own thing. We have to find out what works for this movie and, you know, honor this story which is a whole new story. That’s why I liked how we kind of just dealt with it right up front first scene, first moment that I appeared on screen. Say something about it and then just move on.
Interviewer: What do you say?
Don: What do I say in the trailer? They showed that I think. Yeah, it’s me. I’m here, deal with it. Let’s move on.
What's important here is that Rhodey says, "It's me." There's nothing said specifically about his appearance, and no other character ever makes a comment about Rhodey looking different. So, even though the change was acknowledged for us, the viewers, there was no change for the characters in the fictional reality. The Iron Man 2 opening scene essentially gives us 2 things:
An out-of-universe acknowledgement that things changed, using in-universe dialogue. But, that dialogue, when taken at face value by the fictional characters, really means nothing more than, "I'm here."
An in-universe acknowledgement that for the fictional characters, nothing has changed, because Rhodey is still Rhodey, not new-face Rhodey.
I'm sorry to rely so heavily on the Cheadle/Rhodes scene, instead of anything with Ruffalo/Banner. However, The Incredible Hulk was designed to be both a reboot and a sequel to HULK, so there wasn't a clear continuity there. From The Incredible Hulk to The Avengers, I think the audience is less concerned with the change: The other two movies performed poorly and had a different actor for the title role.
I would note that it's not just Banner that has changed, but his Hulk appearance as well. Not only had he changed facial and body structure, but size and color.
In HULK, he reached about 20 ft in height. In The Incredible Hulk, he's about 9 ft while fully erect. In The Avengers, he's around 7.5-8 ft tall, although we never seen him fully upright. Here are the relevant images presented in the same order:
Here are pictures illustrating his color and other physical difference, again in the same order:
Since these changes are as substantial as changing the actor, it also presents another opportunity for an in-universe explanation. However, there isn't one provided. In this case, I would feel an explanation is more plausible, considering Hulk is a type of mutation, and even in the comics he's had many different appearances. Since the opportunity for explanation was there, but not taken, I believe it's further evidence that there needs to be no in-universe explanation, because in-universe nothing has changed for the characters.
Historically, it's very common for one character to be played by a variety of actors. Looking at plays, for example, where the actor playing lead may change from show to show. There's no reason for an in-universe explanation to break the fourth wall, as they did in Iron Man 2.
Very rarely do movies or shows come up with an in-universe explanation to explain the change in actor's appearances. Famous examples are The Oracle, from the Matrix trilogy, and the Doctor from Doctor Who. However, both of their fictional universes not only have plausible explanations for such a change, but the change is meant to be part of the story.
When it's simply a recasting, the new actor's appearance is not part of the story, and we ignore the change. It's one of the more basic parts of suspension of disbelief.