Why are my gums still so puffy after braces (3 years) with pic?

Your gums look swollen, and that is not a good thing. I can't understand why your dentist would say they're healthy, because swollen gums is a sign of periodontal disease. If you have not been to the dentist in years, then make an appointment asap. If you've been to the dentist within the past six months, and he saw nothing wrong, then you should find a new dentist.