Learn Swedish: Vi stannar här medan du hjälper den skadade mannen. - We are staying here while you help the injured man.

Because medan (while) introduces a subordinate clause. In Swedish subordinate clauses, the subject (du) still comes right before the verb (hjälper). You don't swap them like you sometimes do in main clauses.

Why are we using den skadade mannen instead of just skadade mannen?

This is called double definiteness. Whenever a definite noun (mannen) is described by an adjective (skadade), Swedish requires you to also add the freestanding definite article before it. For a singular en word, that article is den.

Why does the adjective skadad end with an -a here?

When an adjective is used in a definite phrase (following den, det, or de), it must take the definite ending, which is almost always -a. So skadad (injured) becomes skadade.

How do we know Vi stannar means "We are staying" and not just "We stay"?

Swedish doesn't have a separate continuous tense like English. The present tense stannar covers both "stay" and "are staying". You rely on the context of the sentence to decide which sounds most natural in English.