How to explain to an ESL student the use and grammar behind 'very' and 'really'

You've identified the grammatical difference (that really can modify a verb as well as an adjective or adverb): to modify a verb, the equivalent to very is very much.

There is a rather subtle semantic difference as well, at least in the case of adjectives/adverbs.

It seems to me that with an adjective, really implies that the degree is somehow unexpected or beyond the norm. So very big is neutral (it's big, but it might or might not be within the expected range of sizes), whereas really big suggests that it's bigger than such things usually are. (This is only an implication, not a definite meaning).

However, I don't find this implication when really is used with a verb. I really enjoyed it is hardly different from I very much enjoyed it - a little less formal, but no difference in meaning.

[There's also another meaning of really where it is not an intensifier, but means in reality, but that is not what the question was about].

I believe "really" is appropriate with most verbs, adjectives, adverbs. I really like you. It's a really hot day. He runs really fast.

"Very" is appropriate (and means pretty much the same thing as "really") with adjectives and adverbs, but doesn't work with verbs. It's a very hot day (OK). He runs very fast (OK). I very like you (NOT OK).