Why does a lone pair of electrons occupy more space on an atom?

It doesn't exactly occupy more space, but it has a different shape to a bond pair. In a bond pair we have two positive nuclei, with most of the density of the bonding electron pair between the atoms. The outer nucleus attracts the bond pair outwards from the central atom. In a lone pair there is only the central atom to attract the electrons, so they are pulled in more than the bond pair, producing a fatter, squatter shape. This means that more of the electron density is near the central atom than with a bond pair, which makes it more effective at repelling the other electron pairs. Thus there is a difference in the amount of repulsion between different sorts of pair, meaning that he angles between them are different too, in the order, from greatest to least, lone pair-lone pair, lone pair-bond pair, bond pair-bond pair.