How does " The Fruit Battery" work?
To understand how the fruit battery works you need to understand electrochemical reactions and galvanic cells. There's also a nice Wikipedia article describing the Lemon battery.

The basic idea is that the lemon is acidic, meaning there is a higher concentration of hydrogen ions (H+) in solution than hydroxide ions (OH-). When the Zinc is placed into the acidic solution it will tend to oxidize/dissolve into the solution, releasing electrons which pair up with the hydrogen ions to form hydrogen gas. If you don't hook up a wire you're left with a standard galvanic corrosion process because the electrons will simply flow through the solution. However, the acidic/ionic solution is not very conductive compared to a copper wire hooked up and connected to a copper plate, also placed in the solution. The electrons will flow through the wire towards the copper plate, giving it a slight negative charge, which in turn attracts the positively charged hydrogen ions. I seem to remember this increases the reaction speed, but I'm not positive about this (pun intended). The most noticeable effect is that the hydrogen bubbles will be produced around the copper rather than around the zinc, which tends to be saturated with positively charged zinc ions.

The metals don't need to be zinc and copper, and the solution doesn't need to be acidic, though these are often used for demonstration purposes because they're relatively easy/cheap to obtain and safe.

Incidentally this same process is sometimes used to protect metallic components. Ever seen galvanized nails/screws at the hardware store? These operate on a similar galvanic cell where the nail is coated with a zinc layer. As the nail is exposed to humidity/water the zinc will slowly dissolve into solution. The steel/iron core (as well as the copper) will not dissolve because it has a more positive standard electrode potential than the zinc. The zinc in this case is known as a "sacrificial anode" because it's being sacrificed to protect the other metal. It's not even necessary for the sacrificial anode to completely coat the hardware it's protecting, the only requirement is that there needs to be some electrically conductive path between the sacrificial anode and the element being protected. This usually works better than protective coatings which are ineffective when there is a small crack or chip in the coating.