I was recently watching となりのトトロ and came across this. What does it mean? I
feel like I do somehow understand the grammar in here, but as a whole, it
doesn't make much sense to me.

For more context, dad and his daughter are talking on the phone, and the dad
says this sentence to her. So my reasoning is "Get (someone) to let you stay
here"? I don't really know. Or, "Give yourself the permission to stay here"? -
Does that really make sense in this context?
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grammar
What does そこで待たせてもらいなさい mean?
I think you are close.

First consider

  * そこで待たせる = let/make (somebody) wait there.

The sentence is appended by X(し)てもらう, which means _have (somebody) do X_ with
implied benefits to the receiver. So literally

  * そこで待たせてもらう = have (somebody) let (somebody2) wait there.

Now by context, somebody = the people of the house where the girl is calling
and somebody2 = the girl. Thus a literal translation would be: _Have them let
you wait there_. Basically it says "Ask them a favor and wait there".
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I would, respectfully, add a caveat to sundowner's answer.

> X(し)てもらう, which means have (somebody) do X with implied benefits to the
> receiver

I would replace that with

> X(し)てもらう, which means have (somebody) do X with implied benefits to the
> _person talking_ (or someone in their group)

It is a very important nuance for a learner. For instance,

×あんな不良はここで待たせてもらったなんて、許されない。×

is weird, I don't see a context where it would mean something.

！あんな不良はここで待たせてもらったなんて、信じられない。！

is weird too. But one could imagine an heart-felt apology, where the no-good
person is the talker's son, and the talker is very mad at his son and very
impressed with the listener's indulgence. "I can't believe you would let this
no-good son of mine wait here !"

｛息子・僕｝はそんなに豪華な所で待たせてもらって、信じられない！is natural. "I can't believe {I・my son} got to
wait in such a beautiful place!"

It is also important to note that the causative+もらう form can be used more
generally for polite/indirect requests/thanks.

ここで待たせてもらえますか。 "Can I please wait here?" ここで待たせてもらえて、ありがとう。 "Thank you for
letting waiting me here."
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