- We were hot because we'd run.
I am trying to solve this mistake, but it sounds right.
This mistake is grammatical.

- We were hot because we'd run.
I am trying to solve this mistake, but it sounds right.
This mistake is grammatical.

There is no grammatical mistake here. "We'd" can stand for "we had", "we would" or "we should".
In this case, it stands for "we had": "We were hot because we had run."
Just a guess here, but I would say: We were hot because we'd been running. (as has been noted by a previous answer, the we'd is a common contraction that native speakers use to express we had). But "had run" seems awkward compared to my version. (past perfect continuous)
We were hot because we'd run.
The sentence is correct but most English tests and grammar books explain this mistake this way: there should be an action in the progressive aspect preceding the result.