To invoke Feynman in the interview, it can be rather unsatisfying to invoke virtual particles to explain forces in terms of something else more wonted to you. Virtual particles are wholly internal to the analysed system and so individually do not heed normal conservation laws - they are "off-shell" (i.e. off the light cone defined by $E^2-p^2 c^2 = m^2 c^4$) - but their nett effect yields the conservation of momentum laws, which manifest themselves as "force" (i.e. impulse transfer needed to uphold momentum conservation).
You can think of virtual particles more literally as Feynman liked to do, or you can think of them simply as mathematical terms in a perturbation series. Indeed, witness Barut and Kraus, "Nonperturbative Quantum Electrodynamics: The Lamb Shift"; here the system of one electron, one photon (Maxwell-Dirac coupled system) is solved nonperturbatively, and it gives an infinite series of terms corresponding to virtual pairs too. Here the analysis gives the "right answer" (for the Lamb shift) and the virtual pairs drop out of the Maxwell-Dirac equation description - other than this you can think of them as something not really explicable in other ideas more wonted to you. Similar ideas come from the Dyson series, which can either be thought of as being "made of" the actions of virtual particles, or it can simply be thought of as a standard, rigorous technique for finding the fixed points of certain integro-differential operators when they are contraction mappings: mathematicians call this the Peano-Baker series (see Baake and Schlaegel, "The Peano Baker Series" and many mathematicians highly adept at applying it would never have heard of thinking of its terms as "particles".
Another tack: you can think of the force arising from impulse transfer through the "magnetic field" needed to uphold the conservation of momentum and energy, which itself follows through Noether's theorem from basic symmetries of the World - the time shift and spatial shift invariance of physical laws.
However, I'm sure someone will answer with the handwaving argument as to how the idea of throwing balls between two people can - with a stretch - motivate an attractive force. I myself can't even recall the argument properly - I have always found it so unsatisfying that it never sticks in my head and I would ultimately rather stick with the description of the magnetic force in more abstract terms: we don't expect to understand it in terms more wonted to us.
