Learn Russian: Мы заканчиваем работу быстрее.

Because the action is currently in progress. Заканчиваем is in the present tense, focusing on the ongoing process ("we are finishing"). If you wanted to focus on the completed result in the future ("We will finish"), you would use the perfective verb: Мы закончим работу быстрее.

What is the base form of быстрее and how is it formed?

It comes from the adverb быстро (fast or quickly). To form the comparative degree ("faster") for most adverbs ending in -о, you drop the -о and add the suffix -ее, resulting in быстрее.

Just as a quick refresher, why does the word for "work" end in -у here?

The noun is работа, but it changes to работу because it is in the Accusative case. It acts as the direct object of the sentence—the thing that is being finished.