The translation "Please come and see us" for "私のうちに遊びに来てください" seems to me inadequate and misleading, especially in a textbook.
"私のうち" means "my house(hold)", not "us", and thus "私のうちに遊びに来てください" is much closer to "Please come to my house to hang out" (though perhaps less natural as an English sentence, I'm not quite sure).
But apparently in the unexplained and unknowable context of the particular utterance of "私のうちに遊びに来てください" in the textbook example, according to the translation provided the speaker thinks (that the listener knows) that coming to their house entails seeing "us", whoever else that refers to besides the speaker themselves, and these implications are what they decided to express in the translation, not the words actually said!
They could as well (or as badly?) have given "Please come and see me" as the translation, if they'd had a different arbitrary-chosen context in mind for the sentence.
I do think "Please come to my house to hang out" would have been kinder and more helpful, if not over all better, translation.