From what I gather, he backed down because all his top men's families were being threatened. Prigozhin's family was kept safe, of course, but I don't think they expected Russia to threaten the families of those further down the ladder.
Mercenaries and auxiliaries are useless and dangerous; and if one holds his state based on these arms, he will stand neither firm nor safe; for they are disunited, ambitious and without discipline, unfaithful, valiant before friends, cowardly before enemies
Machiavelli, the Prince
Not to call out Putin's Russia for being amateur hour, but the Prince is probably the first thing to read for an aspiring dictator.
I mean he had to know, right? With guys like Putin if you go hot, you either kill him or he has you killed. There is no in-between. His entire image relies on that. Maybe they had Prigozhin's family or something?