The word “restartable” for C stdlib string functions means the function can return an intermediate answer or error in mid-string, and you can then continue processing the rest of the string. For example, returning a byte count or pointer to where it left off, which you can pass back in. In this case I think the mbstate_t handles that.

I believe it refers to the mbstate_t* function parameter. It handles chunked transcoding, where chunk boundaries can be in the middle of multi-byte characters. You can than carry this state from one chunk to the next.