The Science of "Hangover Cure"
This demonstration has several parts which have slighly different science in them.
To start with, you can see that the oil floats on top of the water and you cannot mix them together. This is because the oil is less dense (so it floats) and immiscible (which means it does not mix with the water). Therefore, the food colouring floats on top of the oil, until it is mixed in by hand.
Nothing much happens until some salt is added. This changes the fizzy water a bit. Fizzy water is just water with carbon dioxide gas dissolved in it. Really fizzy water is saturated with the gas - in other words, no more can be dissolved at all.
However, when the salt is added, it changes the water so that it can "hold" less gas, and so a lot of it comes out and bubbles away. However, because the oil is on top, it stops the gas getting away easily and so you get the odd "volcanic" effect (the food colouring isn't important - it just makes it easier to see what is going on).
When that dies down and the amount of gas dissolved in the fizzy water is stable again, an alka-seltzer is added. This creates a lot more carbon dioxide gas of its own and so you get an even more "bubbly" reaction!
(For anther example of carbon-dioxide effect see the film canister rocket).
This is, of course, not a real hanover cure and would certainly make you feel very sick if you were to drink it!