In "He promised that it would change" - if the "it" is an animal, then it would still be "Slíbil, že se změní" in Czech - that a dog would change, for instance, as the personal pronoun would normally be dropped. "František nás prosil, abychom si psa Fíka nechali. Slíbil, že se změní." – František begged us to keep Fík the dog. He promised that it (-the dog-) would change.
If, on the other hand, "it" refers to a situation, circumstances, i.e. "...that things would change", then we'd definitely say "Slíbil, že se to změní." - no dropping of the subject here.