What is .map() doing in this situation?

The .map() function is calling the Array() function with three arguments, the value of the array element which is 0, the index of that element, also 0, and a reference to the whole array.

So it’s like doing this:

var a = [0];
var index = 0
Array(a[index], index, a);   // create array with three elements

The array returned by Array() then becomes the first element of the array that .map() creates, hence the extra level of nesting in your [[0, 0, [0]]] result.

EDIT regarding your edit: when you say [0].map(String); that results in String() being called with the same three arguments like String(a[index], index, a), but the String() function ignores all but the first argument, whereas Array() uses all supplied arguments.