A property handler is the piece of code that enables you to see the properties about your files as in the picture below. Don't worry about the ambiguity in this sentence; I will be much more specific concerning what component does what in later posts.
Here I've created three .book files. This is a file type I made up last weekend, so clearly Windows cannot know how to figure out all the data on the screen above without help. That is where a property handler comes in. It knows about the properties that apply to each .book file and helps Windows to display this information.
In addition to displaying properties, property handlers let users edit them within the standard Windows user interface:
Property handlers also let users search for files based on the property values. I can search for "Haunting" and Windows will show "Blindness" in the result set. And all this can happen from within the standard Windows user interface.
So, in summary, property handlers help your file types fit better into the Windows search and organize experience. By exposing properties directly in the Windows UI, your users can seamlessly interact with your files without having to open them in applications all the time. When they have property handlers, your file types feel less bolted onto Windows.