This heuristic argument is just noting that in a small box, you can't say for sure that you have just one particle, because the walls of the box will allow the particle to fluctuate into different particle numbers, so the idea of "just one particle" is inconsistent with the idea "I have a box which confines the particles".
It is the box that is creating new particles, because the particle is bouncing off the walls. You can't have a particle bouncing off a wall sharper than the Compton wavelength (the inverse mass in natural units) and guarantee that it stays one particle. The interaction of the particle and the external potential always leads to pair creation.
The reason for this is that particle propagation is always faster than light, because energy is always positive. There are no functions which are zero outside the future lightcone whose Fourier transforms are only nonzero on positive energy. This mathematical fact is explained by Dyson, and popularized in Feynman's 1986 Dirac Lectures--- a particle always must be allowed to travel faster than light.
This means that if you have a wall that deflects the particle, and you put a particle far away, it will start deflecting the particle before the particle has time to get to the wall! But in order to not violate causality, this requires that the wall is already doing deflection of something else when there is no extra particle, and this something-else deflection is the virtual particle-antiparticle stuff that is present in the vacuum. If you measure what's going on at the wall for times less than the time to propagate to the wall, you won't be able to tell whether it is the particle that is reflecting, or the vacuum fluctuations, this is how causality is restored.
The mathematical statement that encodes this is microcausality, that field operators commute at spacelike separation.
The states of a quantum field theory are as follows: you have a collection of different positions or momenta. In each position, you can have 0,1,2,3,4... particles (imagine the positions/momenta are discrete and finite volume) in the case of Bosons, or 0,1 particle in the case of Fermions. When the quantum amplitudes for this state fluctuate over time (as they always do in a Hamiltonian formalism) you say that you have old-fasioned virtual particles. This can be rewritten in a covariant formalism, and then the virtual particles acquire a different relativistically invariant language, but the old-fasioned perturbation theory is the source of the virtual particle picture.
Then the statement that there are virtual particle is the statement that you can have mixtures of states of different particle numbers. This is unavoidable when you have a potential box localizing particles, because a potential that can deflect particles can also pair create.