What do you not understand about the below part of the code?
The whole idea behind classes, and encapsulation, is that one can create a new class that has the same interface, but completely different ways of implementing that functionality, and use that new class in place of the old class. The program that uses the class should now know that the implementation of the class changed.
The class that OP is using is not the one designed for the W5100 chip, but it is supposed to provide the same interface and same responses as the one that is designed to work with the W5100 chip.
The comment is a cover-your-ass thing that says only that you have tested the code with the library designed for the W5100 chip and that you know that it works.
There is nothing in the sketch that is dependent on the W5100 chip. It is only dependent an a specific interface that the library that supports the W5100 chip agrees to implement. Other code supporting other chips can implement the same interface, and your code should work.
The fact that it does not, for the poster, is not a problem with your code or his library. It appears to be either a network problem on OPs end, a router configuration issue, or a PEBKAC issue (that's the most likely cause, in my opinion).