Question

how you can explain / prove particles of matter have space between them

Answer 1

Indeed! Particles do have spaces between them. 
 The particles can be dust sized particles or molecules, or atoms, or sub-atomic ones (like neutrons protons and electrons) or the particles defined in the standard model (quarks, leptons etc). But every cluster of particles have spaces between the particles



(This is a picture of atoms taken by an electron microscope)

One can clearly see the gaps/spaces between the dots (atoms).

Answer 2

We can explain it by an ex.
There is no change in volume on dissolving sugar in water tells us that there are spaces between the particles of water .These spaces accomadate the sugar particles .It neans that the particles in water are not tightly packed thye are somewhat loose having space between them