Think of this as meaning, "To them it is good." Notice that "хорошо" is neuter? So something else is implied; essentially, "им (это) хорошо". The same principle works with:
Им холодно -> To them it is cold. -> They are / feel cold.
Им жарко. -> To them it is it hot. -> They are / feel hot.
«Я хорошо́» means 'I'm good' said by someone who is of the neuter gender. E.g. imagine a fairy tale where a sun is personified and speaks about itself.
In real life, living people don't use neuter pronouns (and it's pretty dehumanising to use neuter pronouns about someone, because neuter is usually used for lifeless things), but feminine «Я хороша́» 'I'm good' and masculine «Я хоро́ш» 'I'm good' sound OK. However, you're more likely to hear them with «ты», e.g. «Ты хороша!»/«Ты хоро́ш!» as a form of praise. (It can also be used sarcastically sometimes.)
«Они хорошо́» doesn't make sense, since они́ is plural, and хорошо́ is either a singular neuter adjective (in its short form), or adverb, or predicative verb. But «они́ хороши́» 'They're good' (as a form of praise) works.
Note that those are short forms of the adjective «хоро́ший». Long forms are used with a slightly different connotation. «Они хороши́» brings up images of someone doing things well, but the long form «Они хоро́шие» brings up images of well-behaved children or someone behaving in a kind and naïve way.
You're right, the really correct English would be to use the adverb 'well' to describe how a person is doing. "I'm good" is colloquial slang (especially American I'd say), but strictly wrong...unless you want to say that you are good at something specific or maybe even taste good :-)