I know that the steel is significantly harder than when air cooled, but why is the quenched steel harder?
When steel is slowly cooled, lots of carbon diffusion takes place because it is not very soluble in steel at room temperature. The carbon is in solution at high temperatures, and is rejected out of the lattice as it cools. And when this happens, the microstructure will consist of ferrite and pearlite, and the lattice structure will be base centered cubic (bcc). If it is cooled fast enough, then the carbon gets trapped in the interstitial sites of the lattice and distorts it to a body centered tetragonal (same as bcc, but elongated in one direction) This elongation strains the lattice and makes it harder. Also, when cooled fast enough the atoms do not have time to diffuse like they normally would and they shear into place. This forms the hard phase of martensite that is desired of heat treated steel. But then it must be tempered back some because it is too brittle.
Quenching results in smaller grains than air cooled. This means more boundaries, which means harder steel.
