Y'all would be surprised how quickly someone can put on weight when that's basically your life. Not to mention, he's obviously got the genes to be able to get that big. The real trick is being able to bulk up without losing athleticism.
I trained at API one summer (college baseball). I was on the low-level Amateur plan and literally transformed my body in 6 weeks (not to mentioned drop tenths of a second off my 60 time). This is the same place that trains guys for the Combine. They do every sort of test you can imagine. After each workout (2 workouts per day, speed training, strength training) they gave each of us personalized recovery shakes. Everyone had different amounts of protein, simple carbs, complex carbs, fatty acids, Amino Acids, you name it, that was specifically tailored for our needs, not only based on our body composition, but the type of workout we did.
20 lbs (mostly good weight) in 5 weeks for a guy like Sweazy might seem like a lot, but don't forget the resources/time these guys have. Again, when your singular focus is on training/nutrition, it's a lot easier to accomplish these types of things. Not to mention, the grind of an NFL season can make one lose a lot of good muscle mass. Combine that with taking some time off after the Super Bowl to let the body heal, and it's pretty easy to see how a guy like him good make a big change like that. And to beat a dead horse, the genes to do so helps.