Learn French: Au fur et à mesure, il comprenait mieux.

This is a very common French idiom that means "gradually", "step by step", or "as we go along". The word mesure refers to measurement or pace, so the phrase evokes moving forward at a steady pace. You can use it on its own, like in this sentence, or add que to connect it to an action, like au fur et à mesure qu'il lisait (as he read).

The lesson is about progressive comparisons like "de plus en plus". Why aren't we using that here?

The phrase au fur et à mesure already gives the sentence a strong sense of "more and more" or "progressively". Because of this, simply saying mieux (better) is enough to mean that his understanding was progressively improving over time.

How would I rewrite this to explicitly say "He understood better and better"?

Instead of using au fur et à mesure, you could use the progressive comparison for mieux, which is de mieux en mieux. The sentence would then be: Il comprenait de mieux en mieux. Note that you never say de plus en plus bien, because plus bien doesn't exist in French (just like we say "better" in English, not "more well").

Why is "comprenait" in the imparfait (imperfect) tense rather than the passé composé?

The imparfait is used for ongoing, continuous actions in the past. Since his understanding happened "gradually" (au fur et à mesure) rather than all at once, the imparfait is the perfect tense to show that this was an unfolding process happening over time.