Let's just leap right into this one. Take some styrofoam. Take some acetone- nail polish remover, perhaps, though the video about to be shown uses some more industrial stuff from Ace Hardware. Apply acetone to styrofoam.
Voila, you now have melted styrofoam.
As you can see, you can melt a lot of styrofoam like that. Anne Marie Helmenstine, writing for About.com, estimates that one cup of acetone is enough to melt a bean bag's worth of styrofoam peanuts- remember, styrofoam has 'foam' right there in the name, and all the air in the foam is released when it's melted down.
It doesn't stay melted, though. Styrofoam is a brand name for polystyrene, a hard styrene-based polymer. What the acetone is doing is loosening up the polymer chains, allowing them to move around more freely. When they move more freely, the styrofoam softens, releases the air, and melts. When the acetone goes away, though, the chains re-establish rigidity, un-melting the styrofoam. With the air pockets now gone, you end up with a piece of hard plastic.
Household Hacker, whom I've come to understand have had their videos busted by the Mythbusters on two separate occasions- the online-only 'iOnion' and 'Homemade Surround Sound', has a video of the hardening... right after the ad for Carbonite and before the monologue for Carbonite, plus the actual hardened product barely gets any screentime.
So you get this much-lower-production-quality-but-more-informative video instead. (I will say, though, that Household Hacker suggests letting the melted styrofoam sit for 8-10 hours, which is plenty more than this guy has let it sit.)