Because I don’t think my explanation was amazing, I’ll try illustrating it with a set of three molecules that each have central atoms with 4 electron pairs in their outer shells:
CH4 (4 bonded pairs, 0 lone pairs, angle = 109.5°)
NH3 (3 bonded pairs, 1 lone pair, angle = 107°)
H2O (2 bonded pairs, 2 lone pairs, angle = 104.5°)
Visibly, the replacement of one bonded pair with a lone pair causes a drop in the bond angle due to it repelling more than the bonding pairs, but the replacement of a second results in a further drop. This drop upon the second replacement is in large part caused by the lone pair-lone pair repulsion, since the lone pairs have to separate from each other more than they must separate from the bonding pairs. If the lone pair-lone pair repulsion had no effect, the decrease in bond angle wouldn’t be as significant.