It's basically the former: the radiation given off by the atoms breaking apart triggers more fission when it bumps into other unstable atoms, which just keeps compounding if you have a bunch of fuel in one place until it gets REALLY hot and melts.
To prevent the meltdown you put control rods throughout the fuel which absorb the excess radiation and keep the fuel at useful temperature, roughly like how a fuse is built to burn at a consistent speed rather than simply exploding.