We insist that he uphold the law

Obviously, there is a difference in meaning between "We insist that he uphold the law" and "We insist that he upholds the law" (the first is stating more of a command and the other stating a belief about the situation), but I cannot figure out exactly why the first one is correct, or how to describe to a non-native speaker exactly why you use the plural form of uphold in the first example despite the singular pronoun.

Aside from being the direct object of insist, what kind of phrase/clause is "that he uphold(s) the law" in each case?

how to describe to a non-native speaker exactly why ...

As described above, it's not a plural.

That said, the description you're looking for should involve the contrast between what is a fact and what is envisioned.

We insist that he upholds the law makes his upholding the law a fact.  It is a fact that he upholds the law, and we firmly believe we are right to claim that it is a fact (insist that it's a fact).

We insist that he uphold the law makes his upholding the law an envisioned situation.  We don't claim his upholding the law to be a fact.  We envision him upholding the law, that is, we take the idea of his upholding the law into consideration, and we forcefully declare our desire (insist) that what we envision should become a fact in the course of time.

Other examples you might use:

Mother insists that Johnny [takes / take] his medicine.
The princess insists that no one [comes / come] to see her on Thursdays.
The conductor insists that the musicians [are / be] on time for every concert.

The last one shows quite clearly that singularity and plurality is not the issue in these constructions.

CJ

add one more thing for you:

if the pronounce were we , you should say: We insist that we are uphold the law, instead of saying " we insist that we would uphold the law.

You might know it, but it's just a addtion to the replies.