Learn Russian: Учитель так хорошо объяснил правило, что я всё понял. - The teacher explained the rule so well that I understood everything.

We use так (so) because it modifies the adverb хорошо (well). You use так with adverbs and verbs. If you wanted to describe a noun, like saying "such a good teacher", you would use the adjective form такой.

Is the comma before что required?

Yes! In Russian, you must always put a comma before words like что when they introduce a new clause (a part of the sentence with its own subject and verb, like я всё понял).

Why doesn't the ending of правило change, since it is the direct object?

The word правило (rule) is indeed in the accusative case here because it is the object being explained. However, because it is a neuter inanimate noun, its accusative form is exactly the same as its dictionary form.

Why do we use the perfective verbs объяснил and понял instead of their imperfective partners?

Both actions are viewed as completed events with a successful result. The teacher finished the explanation, which led to the completed result of the speaker grasping the information.

Could I use все instead of всё in this sentence?