I've been asked about this a bunch of times. It works, but I still prefer a real shoeshine. Still, by popular demand...
The image says it all, but to spell it out:
Step one: peel banana.
Step two (and this is KEY): remove stringy stuff from inside of peel.
Step two: wipe on shoes.
Step three: impress everyone with how neat and spiffy you look.
Here's a better tutorial on Wikihow, which says that the reason the peels do such a good job is that the fruit's natural oils work genuine-polish-like magic on shoes; in addition, bananas contain potassium, which, Wikihow says, are "a key ingredient in commercial shoe polish."
I have to call shenanigans on Wikihow. Yes, they do teach you how to polish your shoes with a peel. Yes, the goofy guy picture (below) from their tutorial rules. But potassium? First of all, I can't find any reference to shoe polish that shoes it as a "key ingredient" in shoe polish (Wikipedia lists the product's components as "natural and synthetic materials, including naphtha, turpentine, dyes, and gum arabic.") Second, even if potassium is an ingredient in shoe polish, what would that have to do with anything? The nutrient is in the fruit, not the peel.
Still, the goofy guy rules. Now, go out and shine your shoes.
http://www.bananabook.org/