The answer is: we don't know. However, based on a quick literature search there are two possibilities for speculation. The first is that this is a placebo effect. I have had warts and have tried countless remedies. Yet, odd as it may sound, the one that worked the best was simply to will the warts away. I found that if I seriously concentrated my thoughts and desires to have the wart vanish, it would eventually go away. After that, I never had warts again. I think the treatments that work best are those that convince people to will their warts away. This is also known as the placebo effect.
If you don't buy that, then there are scientific papers that attempt to analyse the components of banana peel but they do not relate this to wart removal. In one 2005 paper by scientists in Japan (no longer online) they list succinic acid, 12-hydroxystearic acid, and a few other organic acids occurring in the peel. Since one of the main components in commercially available wart removal remedies is salycilic acid, also an organic acid, perhaps it is these acids that are the active ingredient. This is just a guess and no experimental evidence exists for this claim. The authors say these compounds have antimicrobial activity. In addition the banana peel contains sugars such as glycoside and monosaccharide components that display significant antioxidant properties. Maybe this helps remove the warts. The bottom line is: we don't know.
Thank you very much for the prompt and very intersting reply. I will be following up on some research on the items mentioned. I have tried the banana peel wart removal for myself...it does work...you can actually see it drying out the wart, which eventually falls off. Only thing is you do have to rub it on at least every second day. For anyone wanting to try this I would be very interested in how it worked/didn't work for you. Method: Take freshly peeled banana skin and rub inside of skin on wart liberally, allow to dry. Repeat every 2-3 days. Wart should disappear within 1 month. Placebo or active compound? You decide.