What does Hawking mean by "Light rays that form the edge of the event horizon could never approach one another"?
Raychaudhury's equation was obtained using parameters involving the propagation of null congruence (a group of light rays spaced out over space and propagating in time) and is used to calculate what are called optical scalars in General Relativity. It is best derived in the Newman Penrose formalism, and one of the scalar parameters included are the congruence 'expansion'. There are two others, one is the shear and the other the twist.
These are described and the equations covered at Wikipedia at https://en.m.wikipedia.org/wiki/Optical_scalars
It was used by Penrose, Hawking and others to prove various theorems on the inevitability of the formation of Black Holes, with horizons, and some properties of any such horizon. One of the results was that to have an event horizon form as collapse proceeds, it is necessary (and I can't remember what else was required to make it also sufficient, perhaps it was the positive energy condition but I'm not claiming I am stating it all correctly) that there are null congruences that are converging, which was defined as the expansion optical scalar being negative. See the reference for the conditions. The
I have not read in full detail Hawking's "A Brief History of Time", but in GR that was used to prove some of the Black Hole (BH) theorems that once a trapped surface forms (and again see the reference for any other conditions), which requires negative expansion null congruences, a BH will inevitably form, and under some conditions lead to the singularity inside the horizon. And it was also shown that at (or near?) the horizon the congruences can ONLY diverge or stay parallel (positive or zero expansion scalar) which means the horizon expands or stays the same, never gets smaller.
That would explain Hawking's claim about the horizon never shrinks. He was just doing it in a popular book, not a scientific paper. Knowing some of the real physics you can understand his non-scientific explanations. I do not know if otherwise they are convincing, they would not convince me but titillate my imagination. That is also why this was a great question, question the popularly stated explanations if you feel you really want to understand. You will never learn, or really understand real physics from popular books, but they are a lot of fun if you know some of the physics, and can give you some intuition, but careful not to do it w/o knowing some of the physics or you'll develop the wrong intuition
See one reference at Wikipedia for the Penrose-Hawking singularity theorems, it talks about converging light rays and it also discusses some the Raychaudhury's equation, which also involved Sachs and other physicists of the time. There's more to read to get it good enough, but it's a start. See https://en.m.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems