Thinking in terms of quantum fields interactions is very fundamental and more intutive than thinking in terms of particle interactions.
Two electrons interaction:
The vacuum is filled with electron and EM field fluctuating at zero vacuum expectation value(also other quantum fields except Higgs).These two electrons are nothing but localized excitations of elctron field at there respective spacetime points. This electron field due to these localized excitations create disturbances in the EM field, causing excitations in the EMF. These excitations in the EMF then travell to the opposite point to transfer the momenta & energy to the localized excitations of electron field, causing the overall effect of electron-electron repulsion.
Because the intermediate particles("photons" in this case) have no overall existence, we say these particles to be "Virtual Particles" (i.e, "virtual photon" in this case).