$self has little to do with $, which is an alias for jQuery in this case. Some people prefer to put a dollar sign together with the variable to make a distinction between regular vars and jQuery objects.
example:
var self = 'some string';
var $self = 'another string';
These are declared as two different variables. It's like putting underscore before private variables.
A somewhat popular pattern is:
var foo = 'some string';
var $foo = $('.foo');
That way, you know $foo is a cached jQuery object later on in the code.