We don't normally think of steel as being flammable, but we are so mistaken: Steel is actually burning all the time. Or at least, it's oxidizing all the time, which is essentially burning, to chemists. We only think of steel's oxidization as "rusting" rather than "burning" because it's happening in slow motion, rather than all at once, as a match would.
Steel wool is a bit different than ordinary steel, though. While a lump of steel has very little contact with the air (just the surface), steel wool has air all up in its business, giving every bit of it access to oxygen. Basically, a lump of steel is to steel wool, what a log is to dried straw in a toaster oven.
NOTE: Cracked is not telling you to put dry straw in your toaster.
Now, because a nine-volt battery has both terminals on one side, touching it to a piece of steel creates a short circuit. Electrons flow from the positive terminal, through the very short segment of wiry steel wool to the negative terminal.
It even works if it's soaking wet.
The steel offers resistance to the electrons and, having nothing else to do with that energy, the whole thing burns. The wiriness of the steel wool means doesn't dissipate the heat, so it quickly reaches critical levels.
Hot steel oxidizes far faster, which in turn produces more heat and, because there's plenty of oxygen around the straw-like steel, that leads to more oxidization. This process spins out of control in the blink of an eye and the heat surges through the steel wool like a 1000-degree wildfire.
Yes, just touching two common household supplies together for a second can cause a fire so hot it'll melt through most anything around you. So definitely don't do this:
No matter how dope it looks.