A good way to control the length of rat claws is to give them something concrete to play on. A half of a cinder block, or a concrete bird perch will work. When they climb and play on it, it blunts their claws naturally and doesn't scare or terrorize them as clipping would do.
Unless they're bleeding, they're probably just scared and uncomfortable. I doubt they're really hurting, but they might be frightened, because they won't understand what you're doing and why. All they'll really understand is being held down to do something unpleasant that they'd rather not do.
If the claws are really bothering you, you could probably nip just the very tip off of the claws with some clippers, but only enough to blunt the point. Otherwise, I would just deal with it and get the concrete block for them to play with. I'm going to have to get one soon, myself, 'cause my boys' claws are getting pretty long, too.
I used to work at a cat shelter so im pretty good with nail clippers. I cut my buck's nail's once every 2 weeks. If you look closely and you have a very calm rat you can cut their nails without hitting the quick. But if you've never clipped rats' nails, i suggest you dont do it until your taught. So, like everybody said, a nail file/concrete toys is perfect.
If the claws are clear (do any rats actually have dark claws?), you should be able to see the quick - the blood supply your veterinarian was speaking of. If you can hold them still enough to confidently avoid cutting into it, this is often less stressful than a file simply because it's over so much more quickly.
BUT -- and this is a big but -- if you're a caffeine junky or otherwise have a natural tremor, don't attempt it. They're too small, and most nail clippers could easily sever an entire toe. Same goes if they're just too wriggly. It's better safe than sorry, and a concrete perch is usually effective.
Edit: Perch, ha! Mind drifted to parrot care, I guess. You know what I mean.