。
She invited Tom and me to the party.
Literal
She [topic-は] Tom [and-と] me [object-を] party [to-に] invite-did-the-favor-of.
~てくれる explicitly frames the invitation as a favor done to the speaker (or someone the speaker counts as in-group). English 'she invited me' is neutral, but Japanese has to mark whose side the kindness flows to — leaving out くれる would make the sentence feel oddly distant, as if the invitation went unappreciated. 招く is the more written counterpart to 呼ぶ in this 'invite' sense.