About the rank check
The rank check picks one puzzle from each of five levels of the 65 endgame puzzles, by empty squares: 10, 11, 12, 13 and 14–15. The engine has read every one of them to the end (exact search) and checked that exactly one move avoids losing (it wins or draws).
Points and ranks
A right answer is worth 2 points, and 1 more if it took 10 seconds or less, so 5 puzzles make 15 points. Two thirds of your points, rounded, places you on the eleven ranks from 10 kyu (0) to 5 dan (10). All right and fast is 5 dan level; all right but slow is about 2 dan. 5 dan level needs every answer right: miss one and it stops at 4 dan.
Titles and remarks
Your title comes from the theme most common in the puzzles you got right (corners, parity, openness, mobility or reading), and the remark from the theme most common in the ones you missed. If time ran out on half or more, the remark says so.
The check is only an estimate. For your real rank, play a full game in the rank exam. Any puzzle you missed can be tried again on its own page, with the explanation.