There could be many reasons. One might be that he's speaking too fast and accidentally swallows air in the wrong way, which leads to gagging. Also, if he has some sort of medical condition that affects his throat or swallowing reflex, that could be a factor. However, it could also be a part of his comedic style. He might be deliberately doing it to create a unique and memorable performance while telling the story, knowing that it will get a reaction from the viewers.
It's possible that there's something physical causing it. For instance, he might have a bit of a cough or some irritation in his throat. Or perhaps he's trying to imitate a certain character or situation in the story that involves gagging, but it ends up looking a bit too real. Another reason could be that he's trying to be overly dramatic in his storytelling and it backfires a bit, resulting in an actual gag.
He could be nervous. Maybe he's not used to being in front of the camera or telling the story publicly, and that makes him gag.