I don't know much about this, but it does seem to me that it's entirely about branding. You sell a school, it requires clever marketing. At this point, Oxford, Cambridge, UCL, King's, Durham, Manchester, they are brand names. It helps that these schools have storied history's, prestigious alumni, etc. They do not beggar themselves out to you, rather, it is the other way around. This is probably why you don't see the best schools in clearing, perhaps even if they do have open spaces. They would rather seem oversubscribed, in this way the mythos and the prestige surrounding these institutions grow and pique the interests of new, potential students.
So, if a smaller school wanted to grow and get bigger and eventually move up the league tables, as someone said before me the research in particular would have to get better. Better research leads to accolades, perhaps not necessarily nobel prizes, but accolades. Accolades lead, if nothing else, to the expansion of a school's global reputation. If I had to guess, this is how newer schools like Warwick and Manchester got to be as well-renowned as they are. Research also leads to a bigger endowment which allows for more expansive recruiting, the hiring of a greater, better number of staff, etc. The school's need to increase their grade requirements. Also, aesthetics play a big role, visual presentation is big. So, everything from the school's website, to the academic buildings and the dorms should be pleasing to the eye. And there's more to it, but you get the basic idea. I think the crux of it all is branding and how well a University can sell itself nowadays.