When a photon is absorbed by tissue, whether by the electrons or the nuclei of atoms, its energy is transferred to the atom. The increased energy causes the atom to vibrate more quickly, which results in the generation of heat. This, of course, is how the microwave oven works. It can selectively increase blood flow to carry excess heat to the surface of the body, where is it radiated into the air, often with the aid of sweat. If the heating is very localized, the heat sensors in the brain may not know that a temperature increase is occurring.
Cell function is very dependent on the ambient temperature.
A complex set of chemical reactions keeps a cell alive. Chemical reactions depend on heat and a change in the expected temperature can disrupt the cell's operation to the point of killing it. While this is a very high temperature for the body to reach, local heating due to the focussing of RF waves in tissue can happen, causing a small area to rise in temperature while the core body temperature changes very little. Preventing this type of RF heating is what the FCC permitted levels of exposure have been developed for. While we can't say for sure either way, we can hypothesize mechanisms that might cause this effect.