You can use the Double Slit equation to prove that a smaller slit width results in a bigger fringe separation but these diagrams are a more "physics" way of doing it. Pick one wave a count the number of other waves that it crosses. The further apart the slits, the more waves it crosses so there are more places for constructive interference and thus more fringes in the same space. The closer the slits, the fewer times a wave overlaps another wave, so the directions of constructive interference are more spread out.
