Why would water rise if the height of the capillary tube above the water surface is decreased?

Suppose the height of capillary tube initially was sufficiently high and water rises to a height h and while it is submerged a part of it is cut off including a portion of water in capillary of height d. Then portion of tube underwater is unchanged but now water rises only to a height h-d as that is the length of tube above the surface of water. As capillary rise is due to surface tension water does not overflow or come out of the tube like a fountain.

I wonder whether the book or other source you are using has expressed the situation unclearly. You are right that the amount of tube above $h$ does not affect the capillary rise. The height $h$ is determined by the adhesion to the surface (and by any ambient pressure difference above the surfaces if there is one; usually we treat the case where that pressure is equal everywhere so we don't have to worry about that).

My guess is that the person writing the statement had in mind the idea of lowering the capillary tube further into the water (without breaking it or anything like that). In this case the capillary height stays fixed at $h$, and therefore the water is staying at the same height as the walls of the tube move in the downward direction; in this sense the water is `rising up the tube' although in fact it is not moving in the vertical direction, but rather the tube is going down. As I say, this is a guess at the intended meaning of the author. The main point is that the capillary height is independent of the part of the tube that is not involved, i.e. the part above $h$.