>When someone uses it sincerely, what do they mean? What does it look like, as opposed to not boxing your time? What specifically does it solve?

A technical analogy would be depth-first search of a solution space, e.g. in a simple chess AI.

If the search has a cut-off depth = deterministically bounded length of search time = time-boxed

If the search does not a cut-off depth = unbounded (potentially infinite) length of search time = not time-boxed