Hunter Oatman-Stanford tracked down an answer from Cosby show costumer Sarah Lemire:
Lemire says that Cosby often strayed from the script, following his gut if he thought it might get a better laugh. “It was incredible and it came out of nowhere, and the director knew to grab that.” As a result, the show often relied on close-up shots of Cosby to capture such moments of improvised humor. However, tight shots like these caused problems when matching the scenes from two different takes, as a slight difference in costume positioning would become a glaring mistake.
“Usually you don’t do close-ups on TV, and that’s why we started using sweaters,” says Lemire. “As our bodies move around, the clothes are going to shift between the first and second take. If you have a jacket on, and the shirt collar’s in one spot, it’s going to slide off a little on one side or the other, or it might do something else that didn’t match. [Director Jay Sandrich] was a real stickler for things matching, so we just did the sweater thing. I actually sewed his shirts to the sweaters so that nothing moved.”
(Video: Part of Vice’s profile of Koos van den Akker, the Dutch designer of many of the most iconic Billy Cosby sweaters.)