Actually - they were AMF/Roadmaster bikes. Someone else said Huffy, but I'm almost certain that's incorrect (unless Huffy made the bikes for AMF/Roadmaster, but I don't think that's true. AMF had their hands in a lot of industries back then, and even owned Harley Davidson in those days). Then again, there wasn't much difference between Huffy and Roadmaster (or Murray for that matter) -- all were cheap, heavy department store bikes made to pretty similar basic specs, though I don't think any of the companies was directly connected to the others. The ones in the movie, as well as in the actual race, were single speed bikes, probably with a coaster brake rear hub.
The actual "Little 500" runs every year at Indiana University -- it's a major tradition and a big party weekend there. I didn't go to IU, but my high school girlfriend did, and I visited. I recall seeing AMF/Roadmaster on the banners for the Little 500 -- it may have been a sponsorship thing, and they supplied the bikes. And even in the film Breaking Away, there's a moment at the start of the Little 500 where the race official announces "Mount your Roadmaster bicycles!"
That movie is still one of my top 10 favorite films of all time. It was also placed in the top 10 of the AFI's (American Film Institute's) 100 Most Inspirational Films.