Learn Swedish: Får vi leka ute? - Are we allowed to play outside?

To ask a yes/no question in Swedish, you just put the verb first. Since får is the modal verb carrying the tense in this sentence, it moves to the front, right before the subject vi (we).

I thought får meant "get". Why does it mean "allowed to" here?

You are right that får by itself usually means get or receive. However, when it is used as a modal verb (a helper verb) together with a second verb, its meaning changes to may or be allowed to.

Why do we use leka here instead of the present tense leker?

Because it comes after a modal verb. Words like får (may), kan (can), and vill (want to) are always followed by the basic dictionary form of the next verb, which is called the bare infinitive. That is why we use leka instead of leker.

Could we use the verb spela instead of leka?

Probably not in this context. Spela is used for playing structured games, sports, or instruments. Leka is the word for imaginative, unstructured playing, which is exactly what children usually do when they play outside.