Well, that's a quite misleading statement. What you are discussing here is the electron self-energy correction due to quantum fluctuations. The point is that in relativistic QFT you use perturbation theory to calculate physical quantities as a formal power series in the electromagnetic coupling contant \alpha \approx 1/137. At 0th order you start with an "uncharged" non-interacting electron, and then you correct for this doing an expansion in powers of \alpha, which is nicely organized in Feynman diagrams.
Doing the radiation corrections, i.e., Feynman diagrams with loops naively, the corresponding integrals are divergent. E.g., the self-energy contritubtion is divergent, leading to an apprantly infinite mass of the electron, which is of course nonsense. The reason is that we never measure an uncharged non-interacting electron but an electron carrying its charge and with it its electromagnetic field, which has energy and this contributes to the mass of the electron. In classical electrodynamics the energy of the field of a point charge is infinite, because to the diverging electric Coulomb field at the point charge's position.
Now, in quantum electrodynamics the problem is solved by renormalization theory, i.e., you subtract systematically the infinities in the calculation of loop diagrams by lumping them to the unobservable bare quantities of the theory (wave-function renormalization for electrons and photons, the electron mass, and the coupling constant). In this way you adapt the observable quantities, like the electron mass and its charge to the measured physical values. This procedure can be mathematically proven to work to any order of perturbation theory, and it is astoningly successful, providing an accuracy of several significant digits in the agreement between the theoretical calculations and the experimental values of the pertinent quantities (like the magnetic moment of the electron, the lamb shift of the hydrogen atom, etc.).
In some sense Susskind is right: It's the electromagnetic field of the electron, and thus in some sense to "photons", which leads to the renormalization of its mass, but the mass is a parameter we have to put into the theory from experiment. Today, there's no way to predict the masses of the elementary particles in the standard model, but they are all tuned to their experimental values. There is no indication that the mass or charge of an electron are fluctuating, when considered as these fundamental constants of the standard model of elementary particles.