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I always get confused about such phrases involving "of". So, could you tell me what exactly "of" means here? Thanks.
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"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!