I think this refers to the fact that unripe bananas can cause constipation, so the speaker sees them as a remedy to their diarrhea attack. But the speaker isn't really asking for the listener to have bananas sent to them: it's a joke.

A quick web search indicates that eating bananas is considered beneficial in restoring electrolytes and carbohydrates in the event they are lost due to the aforementioned medical problem. So, yes, it seems to be meant literally. I’d never heard of bananas in that context — personally, I would try water with low-sodium salt (i.e. salt with potassium chloride) and sugar. You can look up “rehydration formula” or something like that.

Yes, it can be literally asking for bananas, although it could also be intended at least partly as a joke. As Phil reports, bananas are an excellent dietary source of potassium, which is one of the key electrolytes that can get depleted by severe diarrhea. Before rehydration formulas became commonly available, eating one or two bananas was one of the easier ways to help restore electrolyte balance after experiencing diarrhea, heat exhaustion, or a hangover after a night of drinking too much alcohol.

Native English speaker: I also would wonder what it meant. I suspect Evelyn is correct, but it's not exactly clear.