Out of context, this sentence can have so many meanings;
a) he stayed at his (Own) place
b) He stayed at his (Friends or someone else's) place
c) He remained at his post (Like a soldier)
d) He remained in the position in society, for which he was fit! lol :)
Whatever he did, basically, he didn't move!
DL's translation is a good one, and has all the bases covered :)
Generally, Italians would assume that it is his place because "he" is the subject of the sentence, although you are right in that you cannot tell by the pronoun alone. If you want to say "her" and avoid any ambiguity, you might want to say «al posto di lei», so that an Italian would know that it is someone else's place.
P.S. «posto» is not "place" as in "home" but as in "location."
«rimanere» along with handful of other verbs always take «essere» as the auxiliary verb for the passato prossimo because it has to do with a state of being. «avere» as an auxiliary verb usually indicates a transitive action from the verb; «rimanere» is intransitive, though