My teeth are fine but my gums hurt. What's causing this?

It could be. It could be gum disease, food impaction, underlying tooth problems, hormonal, viral or bacterial infection, etc. See your dentist for a definitive diagnosis and appropriate treatment.
See your dentist. You might have some gingivitis or the beginning of periodontal disease. I recommend you see your dentist to evaluate your mouth and gingiva (gums) and then explain what needs to be done.