In epiosde 43, one of the villain from Vanguard Action Squad of League of Villians prevents another villain from going after Midoriya saying that his action was Stein's personal desire even when Midoriya was on the top priority in the kill list.
The disillusionment that Stain and (some) villains have of the hero society is that too many heroes are only in the business for fame, glory, or money. Supposedly, Stain saved Midoriya from the flying Nomu because he saw that Midoriya exhibited traits of being a true hero and would genuinely risk his life for anyone. Many of Stain's later copycats like Spinner would follow this ideology and also respect Midoriya since the later part of Stain's fight was recorded and streamed (making it somewhat public knowledge that Stain respected Midoriya).
I think it's because of how they notice the likeness of his and All Might's quirk. Stain's reason to kill heroes was because of his past. (Don't want to spoil anything so i'll keep my mouth shut.) Stain himself even said that All Might is the only hero who deserves to live. Whatever reason it is, he had the same reason in mind when he saved Midoriya.