I am sorry if this is too complicated a question to answer simply.
I am interested in understanding the mathematical intution behind why standard deviation and correlation are unitless when the metrics from which they are directly calculated (variance and covariance) both have units attached to them.
Its not really obvious as to why taking the square root of the variance produces the standard deviation and why is it unitless, and why does dividing the covariance by the product of the standard deviations always gives a number between $-1$ and $+1$ and why it is the correlation and why is it unitless.
I am unable to find any texts that offer any simple explanation for this. Hence I need some help understanding it as simply as possible.