I should say something about why there are "quantum corrections". It's because of interactions. A bare electron has a charge, but the fluctuations of the electron field produce virtual electrons and positrons which have a charge too, and then those virtual particles produce their own charged virtual particles, and so on. So the physical charge is the bare charge plus the charges of all the virtual particles, and the virtual particles result from the interaction between the electron field and the photon field.
If you try to naively figure out the physical charge, it comes out as infinite. Imposing a cutoff means that you don't allow yourself to consider infinitely nested sets of virtual particles. So the predicted physical charge won't be infinite, but it will still be very large. Renormalization is a practical philosophy which says, we want to match experiment, so we just assume that the bare charge is whatever it has to be to match the physical charge when the quantum corrections are added. The neat thing about renormalizable field theories is that, once you do this assumption for mass, charge, and maybe a few other fundamental quantities, they become predictive again. If you want to predict something about how five electrons interact with each other, you don't and can't calculate a separate "bare" nonsense prediction which then gets corrected by experiment; once you have correctly renormalized mass and renormalized charge, you now have a functioning model of the physical electron at energy scales of interest, and you can make predictions about interacting electrons using that model of the individual renormalized physical electron.
What this seems to be saying is that there is something fractal about particles, because they are surrounded by these clouds of virtual particles which have their own sub-clouds of second-order virtual particles, and so on. But this fractalness isn't infinitely deep; we expect that quantum gravity provides an objective cutoff to this behavior, and also that at energy scales somewhere short of the Planck scale, new heavy particles show up in the fractal cloud, corresponding to physics beyond the standard model. And the renormalization group is just a way to talk about the charge and mass of a fractal cloud of virtual particles, even when you don't know what the bare mass and bare charge of a single bare particle would be.