Well do you know the fundamental theorem of arithmetic? It states that any integer factors uniquely into a product of primes.
Hence if is such an integer it can be written as a unique product of primes ie. . Where are the exponents of the prime number, these can be any integer greater than or equal to 0. For example etc. or , so no matter what we can assign a variable to the value exponent of any prime factor. Hence in the proof in your textbook here is what they did.
Claim: is irrational
Proof: Suppose it is rational, if supposing this leads to a contradiction, then must be irrational.
So if was rational we could write . This imples or that . Now both have unique prime factorizations, so each will have a certain exponent of in their prime factorization. ie. and . So when we square these values we get and .
Now we have that , if we replace them by the factorizations we get , now combine the exponent of the 2 on the LHS, to get .
So now see what what has happened, we have a 2 numbers being equal to each other but for one the prime factor 2 has an exponent of and for the other it's . Since the factorization is unique we must have the two numbers be equal or that , thus we would have an odd number equaling an even number(contradiction). Or we could say the factorization is not unique and these are 2 different factorizations, which is a contradiction again. And it was this contradiction your book mentioned.