Bananas turn brown faster in the fridge due to ice crystal formation, which ruptures the cells in the banana skin and releases an enzyme called polyphenol oxidase.
This enzyme reacts with oxygen to produce pigments that cause browning. While cooler temperatures might slow the chemical reaction, the cell damage in the fridge accelerates the process overall.
The browning is primarily cosmetic, and the banana is still safe to eat. For optimal storage, bananas are best kept at room temperature.