What he means by "evolution at the cellular level" is that every cell in your body normally performs a delicate dance of differentiation, replication, and death to keep all your organ systems operational--this is why skin and bone heal after injury and your lungs don't outgrow your ribcage. It is somewhat akin to how organisms at the ecosystem level perform a dance of diversification, reproduction, and death in equilibrium with their environs.

Cancer is when certain cells in your body become "greedy" about their own survival without regard to the survival of the organism as a whole, and begin to literally compete with the rest of the cells in your body for resources. Normally there are regulatory switches that tell cells to kill themselves when they become damaged beyond the point of repair, but tumor cells have found a way to bypass these safeguards (either because of chance DNA damage at exactly the right sites, genetic susceptibility to malfunction of the safeguards, viral infection, or some combination of the above). As an example of how tumors compete with your other cells, they usually secrete growth factors to create new blood vessels feeding them at the expense of surrounding tissue.