My greatest usage for them is to extend closed-off 3rd party APIs.
Most of the time, when a software developer is offering an API on Windows these days, they are leaning more and more toward .NET for that extensibility. I like to do this because I prefer to depend on my own methods that I can modify in the future and serve as a global entry point to their API, in the case that they change it.
Previously, when having to do this, and I couldn't inherit the API object because it was sealed or something, I would rely on the Adapter pattern to make my own classes that wrapped up their objects. This is a functional, but rather inelegant solution. Extension methods give you a beautiful way to add more functionality to something that you don't control.
Many other peoples' greatest usage for them is LINQ!
LINQ would not be possible without the extension methods provided to IEnumerable.
The reason why people love them is because they make code more readable.
I have noticed another MAJOR usage of extension methods (myself included) is to make code more readable, and make it appear as if the code to do something belongs where it is supposed to. It also gets rid of the dreaded "Util" static-god-class that I have seen many times over. What looks better... Util.DecimalToFraction(decimal value); or value.ToFraction();? If you're like me, the latter.
Finally, there are those who deem the "static method" as EVIL!
Many 'good programmers' will tell you that you should try to avoid static methods, especially those who use extensive unit testing. Static methods are difficult to test in some cases, but they are not evil if used properly. While extension methods ARE static... they don't look or act like it. This allows you to get those static methods out of your classes, and onto the objects that they really should be attached to.
Regarding performance..
Extension methods are no different than calling a static method, passing the object being extended as a parameter... because that is what the compiler turns it into. The great thing about that is that your code looks clean, it does what you want, and the compiler handles the dirty work for you.
