It's a hard to help you here - to give more details for the whole proof would be too time consuming, and may end up confusing you more. I think you need to ask specific questions about what you're having trouble understanding. For example, do you understand the definition of a partition? Can you write out specific examples of partitions? Do you understand what an equivalence relation is, and what equivalence classes are?

One suggestion I have for you when reading the proof, is to use a specific example as you read the argument, and draw lots of pictures. Pick your favorite equivalence relation, and draw the corresponding partition. Then pick your favorite partition and draw the corresponding equivalence relation. Then read the argument and see how it applies to your specific examples.

Again, if you can ask some specific questions it will be much easier for me to help you understand.