What does the syntax function() { name = function() {} mean in JavaScript?

There are two anonymous functions. The first sets YAHOO.example.XHR_JSON to the value of the second anonymous function. This isn't the same as defining a named function. It's just setting a variable to a value that is another function. It's a way of deferring execution. Maybe a second example would help:

var lightFuse = function() { bomb.actionToTakeIn30Seconds = function() { beep(); explode(); } }

This seems to be the line generating the confusion

YAHOO.example.XHR_JSON = function() { (blah blah blah) }; 

This is assigning a function object to a property XHR_JSON on the property example on YAHOO.

It could have been separated out as follows

var v1 = YAHOO.example
v1.XHR_JSON = function() { ( blah blah blah) };

Sounds like you just needed to define FooBar first...? e.g.

var FooBar = function() { alert("Foo"); }
var OtherFunction = function() {
    this.myFunc = FooBar;
}
var instance = new OtherFunction();
instance.myFunc();

var foo = funvtion(){... code ...};

is pretty much the same as

function foo() { ... code... };

The main idea of the code is that the XHR_JSON assignemnt is done after the "load" event of the page is triggered and all DOM elements are established.