"Wheeling by three" is the cavalry equivalent to "right - turn" in the infantry. Starting from the unit being formed in line, the result is a formation which takes (more or less) exactly the same ground as it did before, just with the front to the flank.
Whereas in the infantry everybody just makes a turn to the right by 90 degrees, due to the unmilitary shape of a horse, in the cavalry every three horsemen have to make a turn of 90 degrees to the flank as a group together, wheeling around the center of the horse in the middle. Three horses together one bside the other, take about as much space in the front as each of them takes in length.
Thus, as an example, 90 horsemen in two ranks (45 men in each rank), wheeling to the right by threes, will end up in a column 6 men wide and 15 ranks deep.