The sensation of stepping up and down is related to the vocal cords being adjusted by the stretcher and closer muscles for the pitches being sung.
As one ascends the scale, there is less length, width, and depth of cords vibrating. The singer has the feel of stepping up and back the vocal staircase, from basement to first floor to second floor to the attic.
As one descends the scale, there is more length, width, and depth of cords vibrating. The singer has the feel of stepping down and in front.
All of this feel is just a reflection of adjustments taking place at the level of the cords. The feel is only present when the function is correct. First, the function is established and improves. Then, one begins to have proper sensations that testify to vocal health.