Literally or naturally?
Literally it utilizes the permissive usage of the causative. As in, to let something happen rather than to necessarily cause it directly like the name “causative” might incorrectly make you think. So in this case, you’ll saying ill have (もらう) you let me (causative) come in. In this case, あがる just means entering so 部屋にあがる is to enter a room or apartment in this case.
Naturally…hmm I hate translating stuff but I guess the most natural sounding thing to say that gets the same feel across might be “I’ll head on in for a bit” as opposed to “I’m coming in for a bit” to maybe make it a bit softer?
we kinda do the same thing in english to some extent. Like if I say “allow me to demonstrate” im really saying “I’ll show you” but in a more roundabout way I guess and im not actually asking for your permission.
No problem! Its a very common one, but it can be tricky if you’re not prepared.