So these bananas have been brought home and are green. How do they turn yellow? Don't they need sugar or something to turn yellow? But where does the sugar come from if the bananas are already picked off the tree?

I don't know about the colour. But they have sugars in them from the start. Green bananas are mostly starch, which is a long polymer of glucose molecules. As they ripen thiis is hydrolyses (broken up) to individual glucose molecules, some of which isomerise to fructose. The glucose and and fructose condense (join together) to form sucrose.

So does the water that is necessary for hydrolysis comes from the tree that the bananas came from? Then the bananas store it for a while or something? Or does the water come from the water vapor in the air?

It would come from the cytosol of the fruit, and whilst it's on the tree this would be topped up by the tree.