Aha, initially I thought the question is asking about a distance between two points, vertices , which is not asked in this question. Cube's diagonal would be $d=side*{\sqrt{3}}$.

Here, surface distance is the distance on the surface of cube's sides. Like, we are drawing a line on the sides of a cube shaped box. Therefore, the required distance is a diagonal between two sides and the side itself $side*{\sqrt{2}}+side$. Answer is $6*{\sqrt{2}}+6$, and it's indeed A.

Thanks for posting a good question.