How can you tell a puppy leg is broken?

You would have to see the vet to find out if it is broken or if there is another problem and for proper treatment. If a broken bone is left to heal on it's own without the vet it can cause permanent damage (often painful as well) as well as infection is a possibility or even an eventual need for amputation depending. An exam is inexpensive and an x-ray will probably be recommended but you can go over all of that once the pup is examined.
It depends on the dog as well as the break but some barely react at all other than some limping and maybe a quick yelp, particularly with a more minor break. Since they have a much higher pain tolerance than we do if they are showing signs of pain a phone call to the vet to discuss whether or not they should be checked out is needed at a minimum or for instructions on what you need to be doing at home and when it means your pup has to go in and things like that. It's always best to at least check (or have them seen) since putting off seeing the vet when needed generally means much higher bills shortly after (or worse).
Your regular vet is best but if finances are a concern you can also check into low cost clinics in the area, maybe attached to your local shelter or aspca, programs in the area and things like that but just an exam to get an opinion is very inexpensive and you can go from there.
Limping can also indicate things like sprains, other injury's, a luxating patella out of socket, torn ligaments and other things so it's important to consult the vet to find out what the problem may be and what your options are and what needs to be done for your pup.
In the mean time restrict activity (kennel rest) so your pup isn't making anything that may be wrong worse by moving around and no furniture, stairs or anything like that. Hope it's nothing much and your baby is feeling better very soon.