Learn Portuguese (Portugal): Eles têm orgulho em nós.

Because it comes immediately after a preposition. In this sentence, that preposition is em. After prepositions, Portuguese uses stressed pronouns. For the first person plural, the stressed pronoun is nós. It looks exactly like the subject pronoun for we, but sounds slightly different from the unstressed object pronoun nos.

The English translation says proud of, but the Portuguese uses têm orgulho. Why?

In Portuguese, instead of using an adjective like proud, you usually say to have pride. The word orgulho is the noun for pride, so Eles têm orgulho literally translates to they have pride.

Why does the sentence use the preposition em instead of de to mean of?

Because the Portuguese expression maps better to the English phrase pride in. While English speakers say proud of, Portuguese speakers use orgulho em to mean pride in. You might also hear ter orgulho de, which is equally correct and means the exact same thing.

Why does têm have a circumflex accent (a little hat) on the e?

The accent tells you that the verb is plural. Eles têm means they have, whereas ele tem without the accent means he has. It is a very important spelling rule to remember for the verb ter, showing you who is doing the action.