Callbacks allows you to insert your own code into another block of code to be executed at another time, that modifies or adds to the behavior of that other block of code to suit your needs. You gain flexibility and customizability while being able to have more maintainable code.
Less hardcode = easier to maintain and change = less time = more business value = awesomeness.
For example, in javascript, using Underscore.js, you could find all even elements in an array like this:
var evens = _.filter([1, 2, 3, 4, 5, 6], function(num){ return num % 2 == 0; });
=> [2, 4, 6]
Example courtesy of Underscore.js: http://documentcloud.github.com/underscore/#filter