** Disclaimer: This is a non technical way to explain extended top hats in relation to shock/damper travel and suspension.**
So, I have seen a number of people ask this very same question, and unfortunately there is little information on this topic. It all seems to be spread over the site. So here is your one stop shop for extended top hat info.
Exhibit A: This is a picture of extended top hats thanks to the Google:
Exhibit B:This is a picture of the extended in red, and oem in black:
If you aren't blind, you can clearly see that the red top hats are "taller" than the black top hats.
The first problem that we run into with top hats and suspension is that people want to lower their cars improperly, or "slam" their cars to the ground for whatever reason. When you "slam" your car, the shock body sits closer to the top hat, giving you less available damper travel, or less room for the shock body to move. If you then hit a good sized bump, your shock body may hit your top hat (hopefully hitting a bumpstop first) causing damage.
That said, the first purpose of Extended top hats is to solve the problem of hitting your top hats with the shock body.
If you look at the above pictures, the bottom picture best shows the difference between the two. The middle part of the extended top hat is "taller", and just happens to be big enough to allow for the shock body to slide into it. That is not coincidence. Because of this extra room, when you hit that "good sized bump" your shock body now has the extra room it needs to travel to not hit the top hat.
Again, when a car is lowered, the shock body will sit closer to the chassis/top hat. Because of this there is less available suspension travel to handle bumps making it more likely to cause the shock body to have impact with the top hat. If you were able to provide an extra inch or 2 of extra shock body travel with the extended top hat, you would effectively increase the room in which you have to work with, and lower the chances of hitting the top hat.
Another reason for extended top hats is to help your shocks/dampers travel within their optimal range. The problem here is that each shock will have an optimal working range usually not known by the general public until you blow out your shocks. Some companies will give you a suggested drop measurement range, let's face it... not everyone listens to that.
The way extended top hats help your shock to travel in their optimal range is by changing the resting point of the shock body in relation to the damper shaft. If you are starting out an inch or two further extended then you would without the extended top hat, you are then giving yourself the proper range in relation to your starting point to work with. How much more room you need depends on your drop and the shock/damper you are using. Unfortunately, there are way too many options to have a solution layed out for each situation. You should just make sure that your shocks/dampers are properly rated for the spring rate and height that you want to achieve.
What extended top hats DoN'T do:
Extended top hats will NOT change the ride height of your vehicle. This is because your car is resting on springs, not solely on your shocks/dampers. The extended top hats are only moving the point at which the damper shaft bolts to the chassis and top hat, pulling the shaft out of the shaft body more, not extending your springs.
Extended top hats will NOT make blown shocks work... if your shocks are blown, they are done for. Replace them with proper shocks/dampers for your setup.
Extended top hats will NOT make OE shocks, or shocks not rated high enough for your application work. Use the right shocks/dampers for your application. Extended top hats will not be the end all solution to a poorly designed suspension setup. They have their purposes, and will only work when used properly.
**once again, this is a NoN TECHNICAL way to try to explain extended top hats to the masses. I know that there is physics behind all of this, but let's keep it there ie: behind all of this.**