As for the fountain itself, before you open the bottle of tonic water the carbon dioxide that makes it fizz is dissolved in the liquid. When you open the bottle, you release the pressure of bottling and some of that carbon dioxide comes out of solution, making your soda bubbly. The bubbles are free to rise, expand, and escape.
When you drop the Mentos candies into the bottle, a few different things happen at once. First, the candies are displacing the tonic water. The carbon dioxide gas naturally wants up and out, which is where it goes, taking some liquid along for the ride. The soda starts to dissolve the candies, putting gum arabic and gelatin into solution. These chemicals can lower the surface tension of the soda, making it easier for bubbles to expand and escape. Also, the surface of the candy becomes pitted, providing sites for bubbles to attach and grow. The reaction is similar to what happens when you add a scoop of ice cream to soda, except much more sudden and spectacular (and less tasty... a lot less tasty).