What is the hidden meaning of electrochemical stability window?
The electrochemical stability window is most important when considering components of an electrochemical system that you do not want to be oxidized or reduced. This refers most often to the electrolyte or protective coatings.

For example, in lithium ion batteries it is highly desirable that the electrolyte does not react/change/degrade in any way as a result of battery charging/discharging or even in storage. A lithium ion battery typically has a voltage of 3.6 V. The electrolytes used are either lithium salts in organic solvents (ethylene carbonate, etc), solid ceramics, or polymerics electrolytes. If the electrolyte was water based, the electrochemical stability window would only be 1.2 V. 

Why 1.2 V?

The standard reduction potential of water is 0.0 V (vs SHE)

The standard oxidation potential of water is 1.2 V (vs SHE)

You can not operate a 3.6 V lithium ion battery with an aqueous electrolyte because the electrolyte itself would react, creating dangerous hydrogen and oxygen gas. So you need an electrolyte with a larger stability window.