My analogy for an explanation relates it to explaining the way to the station: you describe the general landscape on the way, not every crack in the pavement. You are assuming a certain known structure, or type of structure. (Such as: turn left at the traffic lights, then right at the Post office.) You do draw attention to any major holes in the path.

In mathematics we often have to invent the structures which make the ideas clear. I heard Raoul Bott in 1958 say of Grothendieck that he was prepared to work very hard to make a proof tautological. Grothendieck was very keen on understanding the underlying processes and forms.

One importance of this is for computational help: we would like computers to deal with the various hierarchical levels in mathematical practice, and not to have to search for a proof using a low level of structure.