Learn Polish: Mieliśmy dużo pracy, dlatego nie mogliśmy wyjść z biura. - We had a lot of work, that's why we couldn't leave the office.

The verb móc (can / to be able to) has irregular stem changes in the past tense. For groups containing at least one man (masculine personal), the core stem changes from the singular mógł to mogli-. Then, you simply attach the standard "we" ending -śmy to get mogliśmy.

Is the stem change in mieliśmy (from mieć) happening for the exact same reason?

Yes! Both mieć (to have) and móc (to be able to) change their vowels before the -li- ending used for masculine personal groups in the past tense. If the "we" in this sentence was a group of only women, the endings would use -ły- instead, and the vowels wouldn't change in the same way: you would say miałyśmy and mogłyśmy.

Why does the sentence use pracy and biura instead of their dictionary forms, praca and biuro?

Both words are in the Genitive case here, but for different reasons. The word dużo (a lot) always requires the noun following it to be in the Genitive case, turning praca into pracy. Later in the sentence, the preposition z (meaning "out of" or "from") also requires the Genitive case, turning biuro into biura.