When you "go bananas" you behave in a slightly crazy or silly fashion. The expression is considered slang and is therefore used only in informal contexts.

Literally everyone, ever. Including the writer of this definition.

When you find yourself in the midst of an argument on a subject that you have no knowledge of and an obvious polarization between the preferences, you can always pick the side of the venerable banana. It will get you nowhere, but at least you can say you participated.

Taken from the Simpsons episode in which Bart and Nelson race an orange and a grapefruit to the end of the schoolbus. Ralph Wiggum decides to join the race and puts a banana in the aisle and yells "Go banana!"