The starting Explorer is a cruiser, not a fighter. Consequently it cannot be repaired by carriers. It will, however, be repaired by factions if they end the turn in a green square, same as everything else.
That being said, the confusion is understandable because the explorer looks a lot like the basic starting fighter. Something to add to the backlog of stuff; maybe redraw the explorer icon to be more cruiser-ish.
Although as we all know, the brothers are busy making STF the best game they can as their highest priority. I suspect that the next really big update to 4X will be a brand new game on a brand new engine -- if there is enough interest.