I have found that a strong indicator of research ability is a student wanting to know why something is true. There is also an interesting distinction between an explanation and a proof. (I gave up using the word "proof" for first year students of analysis, and changed it to "explanation", a word they could understand. This was after a student complained I gave too many proofs!)
A proof takes place in a certain conceptual landscape, and the clearer and better formed this is the easier it is to be sure the proof is right, rather than a complicated manipulation. So part of the work of a mathematician is to develop these landscapes: Grothendieck was a master of this!
Of course the more professional a person is in an area the nearer an explanation comes to a rigorous proof. But in fact we do not write down all the details. It is more like giving directions to the station and not listing the cracks in the pavement, though warning of dangerous holes.
The search for an explanation is also related to the search for a beautiful proof. So we should not neglect the aesthetic aspect.
