Why do specifically bananas go brown quicker in the fridge?
What's interesting with this one is we don't really know very well the mechanism behind what's called chilling injury. It happens to a range of fruits, like bananas, peaches, avocado, or apples. The belief is that the chilling alters membrane permeability to storage vacuoles inside the plant cells. Try On Food and Cooking, pp.269, and Puig et al. (2015) for references here.

Now the membrane permeability to the storage vacuole changes (it gets leaky), and inside these vacuoles are phenolic compounds. In the cytoplasm of the plant cell, there's an active enzyme called polyphenol oxidase (PO). The reactions it catalyzes end up with a compound called melanin, a brown/black pigment:



Fig A. An overview of what a PO might do.

So the phenolic compounds leak out and PO goes to work catalyzing reactions that start building up the dark pigments associated with browning. Being said, I can't at this moment find a good paper that says "look, this is how the entire process happens," but I'll be attempting to read a little deeper into the matter.

Keywords: flesh browning, enzymatic browning, chilling injury, polyphenol oxidase, storage quality, storage conditions