I always get confused about such phrases involving "of". So, could you tell me what exactly "of" means here? Thanks.
"Very nice of you" means "That thing, which was done by you, was very nice".*

"Very thoughtful of him" means "That thing, which was done by him, was very thoughtful".

And so on.

In full, you need a subject at the start of the sentence (e.g. "That was very nice of you") but often in conversation the subject is dropped and implied by context.

I used the past tense ("was") in the paraphrasing, but these expressions are not tense-specific so "is" or "will be" are just as valid.

One of the many uses of "of". It's sense 2d at : "on the part of" (an interesting definition that incorporates the word it's defining, but we shouldn't be too critical since defining the meanings of prepositions is hard!).

*Edit: I don't know why in the world I didn't write this as "That thing which you did was very nice" etc. Anyway, you get the idea!