Ask the one who wrote it in your notebook to explain it to you! Just kidding.

OK, but do you understand thedefinitions, like ? The idea of the proof is to show that the number of subsets with k elements equals the number of subsets with n - k elements. To do this, the author defines two collections of subsets: one has all subsets with k elements, the other has all subsets with n - k elements. To show that these collections are equinumerous, the proof defines two functions φ and ψ from one collection to the other and claims that they are mutually inverse. That is, each element of the first collection corresponds to one and only one element of the second collection.

I am not sure what it means to understand a function. A definition is a definition: it says, take this, do this operation, call the result thus. There is nothing to understand. You may not understand therationalebehind the definition, but first you need to follow through the proof and see if it works out. You may find that it is a valid proof even if you don't have an intuitive grasp of it.