What exactly makes bananas go brown?
The reason for this is the oxidation of phenol residues in the banana (for example in the yellow color) which get oxidized by the enzyme Polyphenol oxidase (PPO) to melanins. The scheme (the image is from this website on food browning) looks like this (you can of course also use more complicated substrates):



For further information, see these references:


Banana Polyphenoloxidase. Preparation and Properties.
Molecular cloning and characterisation of banana fruit polyphenol
oxidase


Polyphenol oxidases are widely found in fruits and cause most of the oxidative browning. See these articles:


Polyphenol oxidases in plants and fungi: going places? A review.
Enzymatic Browning, Polyphenol Oxidase Activity, and Polyphenols in
Four Apple Cultivars: Dynamics during Fruit Development