There are several true ideas that the people who said that might have been getting at. However, the picture of little tiny dot-like particles fitting in empty spaces is definitely not even close to being a correct way to describe things.
All small-scale things (and maybe large scale things too) are quantum objects. They don't have definite values of position, velocity, sometimes energy, or even number of particles. So 'empty' space has the possibility of acting as if it has particles in it, if something pokes it the right way. That's true for any sort of particles, but the ones that show up most easily are the one with lowest rest mass. The electron/positron pair is particularly important. Neutrinos have even lower rest mass, but they don't interact much with anything. (We're not counting the particles with zero rest mass, the photons, at your request.)
Perhaps what you heard referred to the wave nature of particles, particularly electrons. The electron state is spread out, and a weak tail of it extends far out from each atom. In ordinary materials, there's no region between the atoms where the chance of finding an electron is really zero.
(published on 08/21/09)