How the plate height is assumed to be the standard deviation?
I feel that you misunderstood what the authors were trying to say in that textbook. Forget about everything for the time being.
Imagine if you injected a single analyte band into an HPLC column of length L at a constant flow rate. You waited for some time, and now the band is exiting the column. For simplicity, assume that the band is rectangular.
Ask yourself, how much average distance did the analyte band travel? Indeed, it must have traveled a distance L if it has to exit the column. Most modern HPLC columns are 10 cm long; assume analyte traveled an average of  L=10 cm.
Now a band cannot be infinitely thin inside a chromatographic column. The analyte spreads in space, i.e., its width increases as it passes through the column due to diffusion and other processes.
So let us say the band is now a 0.05 cm wide rectangle when it reached the end of the column. Half of it is inside the column, and half of it has exited the column. What width of the band is inside the column, and what width is outside the column? You would say 0.05/2 cm of the band is inside the column, and 0.05/2 cm of the band is outside the column.

Total distance traveled by the portion of the band which has exited = 10 + 0.05/2 cm
Total distance traveled by the portion of the band which is inside = 10 - 0.05/2 cm

Now in chromatography, as you have seen, peaks are not rectangles, but they are Gaussians. Furthermore, Gaussians are characterized by their &quot;standard deviation,&quot; which is a measure of their spread. If you apply the sample logic

The mean (=center of the Gaussian) has traveled a distance of 10 cm or L
The portion of the Gaussian which is still inside traveled a distance L-1*standard deviation
The portion of the Gaussian which exited the column traveled a distance L+1*standard deviation.

Note that conventionally a full Gaussian band at the base is considered to have a width of 4xstd. dev.