It’s definitely a tricky word. In this specific sentence, “żeby” is absolutely necessary. The standard translation would indeed be “in order to”. However, it doesn’t always translate so easily into English, which is why many learners find it difficult to grasp. Other possible translations include “ to”, “so as, “so that” (according to Wiktionary).
The main function of “żeby” is introducing subordinate clauses of purpose, and this is usually where it is translated into “in order to”:
Robię to, żeby zwyciężyć. (I’m doing this [in order] to win.)
Żeby zarabiać pieniądze, trzeba pracować. (In order to earn money, one has to work.)
“Żeby” is also used after verbs of permission, warning, request, command, desire, concern, hope,
preference. (quote from Oscar E. Swan’s textbook), and “pozwolić” is one of them. Some other examples:
Chcę, żeby wszystko było dobrze. (I want everything to be fine.)
Powiedział mi, żebym zadzwonił jutro. (He told me to call tomorrow.)
Anna prosi nas, żebyśmy pomogli jej mamie. (Anna is asking us to help her mother.)
There are also some other minor uses, but I think these are the two most common ones.
So why do you need to use “żeby” in all these contexts?
Well, Polish doesn’t really have a direct equivalent to the English “to”: a sentence like I want to eat lunch. is simply translated as Chcę zjeść obiad. In this simple example, we don’t need any word to link “chcieć” and the verb, since it is obvious who is the subject of the sentence and what is its object.
However, if you want to say I want you to eat lunch, you can’t simply insert “ciebie” between “chcieć” and the verb, as you do in English. “Chcę ciebie” only makes sense if you literally want the other person. The correct way to say this is Chcę, żebyś zjadł(a) obiad. The purpose of “żeby” here is to make it clear that the person who is supposed to do the “eating” isn’t the same as the person doing the “wanting”.
Note that “żeby” takes the suffix -ś here, which tells us that the clause is about second person singular. If you change it back to the regular “żeby”, it will mean the third person. The first person forms are “żebym” (singular) and “żebyśmy” (plural), and so on.
Of course, this is a simplified explanation. If you need more context, I suggest that you browse the sentences in Clozemaster’s database—a search for “żeby” gives 420 results, that should be enough to get a better idea of how it works in practice.