Why lone pair occupies more space than bond pair?
The basic premise of this statement is that the repulsion exerted by an electron system depends upon the area it occupies. More the area, more is the repulsion.

As the bond pair is formed from the sharing of electrons, it experiences attraction from not one, but two nucleii. On the other hand, the lone pair is attracted by only one nucleus. 

So, the bond pair is more localised while the lone pair is spread out and occupies a larger area.

The 'size' here refers not to the size of electrons but to the size and nature of the orbitals these electrons occupy.