(Rewritten to clarify, simplify)
That statement performs a couple different actions:
myVar = // This portion is a regular assignment, it will store the value of the suffix
!!someOtherVar; // This portion is evaluated to a boolean result
The !!someOtherVar, I assume, is what you're really asking about. The answer is simple: it performs two logical NOT operations against the truthiness (a Javascript'ism) of someOtherVar.
In other words, if you understand the ! operator, this just combines two of them (!! isn't a different operator). By doing this it essentially returns the boolean evaluation of someOtherVar--in other words, it's a cast from whatever type someOtherVar is to boolean.
So... to walk through this, and pay attention to the result of myVar:
myVar = someOtherVar; // myVar will be whatever type someOtherVar is
myVar = !someOtherVar; // myVar will *always be boolean, but the inverse of someOtherVar's truthiness
myVar = !!someOtherVar; // myVar will *always be boolean, and be the equivalent of someOtherVar's truthiness
