First, you must be a licensed physician. That eliminates the chances of quacks and charlatans who might otherwise call themselves sports physicians. Okay, so you're an MD. You've probably specialized in Family Practice, Internal Medicine, or something similar, so you've got a good baseline set of skills. Now you've got to complete an extra year or two of sports medicine training, probably in the form of an accredited fellowship program. Then you get to take a qualifying Sports Medicine subspecialty exam. Assuming you get through that, you can hang out your shingle as a sports physician. You'll need to meet periodic continuing education requirements as well.
Remember that orthopedic surgeons are often called sports physicians, too. If you thought getting your MD credential was difficult and time consuming, wait until you have to complete all that extra surgical training. Once you finish that, you'll likely undertake a surgical sports med fellowship, which tacks on another one to two years of training. You might choose to become well versed in general sports-related orthopedic problems, or you may opt to focus on one body part, such as knees or shoulders. Any way you cut it, you're in some sort of training for years.