Your elbow is a very important but delicate joint that can get aggravated from overuse — and bad-looking pull-ups. Elbow pain sucks, but if you want to keep doing pull-ups for their very real benefits, try a "neutral grip" where your palms face each other. ATHLEAN-X's video above goes over some ways to do pull-ups with less or no elbow pain, including switching to a neutral grip. To explain why the grip makes a difference, you first need to understand that elbow pain from pull-ups likely starts because you're probably trying to cheat your reps.
Think about what you are doing: In a traditional pull-up (palms facing forward), you naturally have to twist your wrist and generally do whatever you can to finish out the rep. When that happens, your elbow is absorbing a brunt of that tension and stress; then disaster strikes as you gradually wear down your elbow joint. With a neutral grip, your wrists don't need to rotate or experience nearly as much wear and tear.
Essentially, don't try to cheat your reps. Get your form right: A good, safe pull-up does not begin at the elbow. The keys are to have your elbows at yours sides (not in front of the body) and start the movement by contracting your lats and bringing your shoulders down.
Elbow Pain with Pull-Ups (QUICK FIX) [ATHLEAN-X]