AngularJS remembers the value and compares it to previous value. This is basic dirty-checking. If there is a change in value, then it fires the change event.
The $apply() method, which is what you call when you are transitioning from a non-AngularJS world into an AngularJS world, calls $digest(). A digest is just plain old dirty-checking. It works on all browsers and is totally predictable.
To contrast dirty-checking (AngularJS) vs change listeners (KnockoutJS and Backbone.js): While dirty-checking may seem simple, and even inefficient (I will address that later), it turns out that it is semantically correct all the time, while change listeners have lots of weird corner cases and need things like dependency tracking to make it more semantically correct. KnockoutJS dependency tracking is a clever feature for a problem which AngularJS does not have.
Issues with change listeners:
Syntax is atrocious, since browsers do not support it natively. Yes, there are proxies, but they are not semantically correct in all cases, and of course no proxies on old browsers. The bottom line is that dirty-checking allows you to do POJO, whereas KnockoutJS and Backbone.js force you to inherit from their classes, and access your data through accessors.
Change coalescence. Suppose you have an array of items. Say you want to add items into an array, as you are looping to add, each time you add you are firing events on change, which is rendering the UI. This is very bad for performance. What you want is to update the UI only once, at the end. The change events are too fine grained.
What about performance?
So it may seem that we are slow, since dirty-checking is inefficient. This is where we need to look at real numbers rather than just have theoretical arguments, but first let's define some constraints.
Slow Anything faster than 50 ms is imperceptible to humans and thus can be considered as "instant".
Limited You can't really show more than about 2000 pieces of information to a human on a single page. Anything more than that is really bad UI, and humans can't process this anyway.
So the real question is this: How many comparisons can you do on a browser in 50 ms? This is a hard question to answer as many factors come into play, but here is a test case: http://jsperf.com/angularjs-digest/6 which creates 10,000 watchers. On a modern browser this takes just under 6 ms. On Internet Explorer 8 it takes about 40 ms. As you can see, this is not an issue even on slow browsers these days. There is a caveat: the comparisons need to be simple to fit into the time limit... Unfortunately it is way too easy to add a slow comparison into AngularJS, so it is easy to build slow applications when you don't know what you are doing. But we hope to have an answer by providing an instrumentation module, which would show you which are the slow comparisons.
It turns out that video games and GPUs use the dirty-checking approach, specifically because it is consistent. As long as they get over the monitor refresh rate (typically 50-60 hz, or every 16.6-20 ms), any performance over that is a waste, so you're better off drawing more stuff, than getting FPS higher.