Why square-root of Hanning Window
The purists will tell you it should be called a Hann window, or VonHann, but not Hanning.

From a math perspective, the Hann window is also equivalent to a sine squared window at half the frequency, so by taking the square root you are getting a sine window.

Why use a window at all?  The seminal paper on the topic is: 

harris, fred. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform"

A link to the paper can be found in reference [10] of the Wikipedia article on windows.

Hope this helps.

Ced