Learn Swedish: Du får inte hoppa i backen! - You must not jump on the slope!

You are right that att få often means "to get". However, when it is used as a helping verb with another verb (like hoppa), it means "to be allowed to" or "may". Therefore, du får inte literally means "you are not allowed to", which is how Swedish expresses "you must not".

Why is the word inte placed between the two verbs?

In a normal main sentence like this, the rule is that inte (not) comes directly after the first, conjugated verb (får). Any other verbs, like the infinitive hoppa (to jump), must wait and come after inte.

Why does the sentence use i (in) instead of på (on) for "on the slope"?

Prepositions are notoriously tricky because they rarely translate directly. In English, you ski or stand "on" a slope, but Swedish uses i (in) for being on a ski slope or hill. So, you are i backen.

What is the base form of the word backen?

The base word is en backe (a hill or a slope). To say "the slope" in the definite form, you just add -n to the end of the word, making it backen.