How does this follow? I understand that PRNGs let you construct randomized algorithms in practice, but how do you transfer probably polynomial runtime to definitely polynomial runtime?
The definition of a PRNG in complexity theory (for the purposes of this topic) is that no polynomial-time algorithm can look at its output and distinguish it from real randomness (with high confidence etc etc).
Now imagine you have a PRNG whose seed size is logarithmic in the output size. You can compose it with our BPP algorithm and iterate over all the possible seeds in polynomial time (because the seed is logarithmic), and take the majority result. You built a polynomial time algorithm that, if it gave incorrect results, could distinguish the PRNG from real randomness. So it must be correct.
It's unknown whether ZPP = P (but it almost certainly does). It is known that ZPP is the base of the random hierarchy... ZPP = RP u co-RP, where RP is the class of languages that are accepted with bounded error.
Any good complexity textbook beyond Sipser will discuss the randomized complexity classes at extreme detail, since they are crucial for constructing all the other interesting complexity classes (notably IP and MIP).
Are there problems in BPP that can defend against "helping" PRNGs by diagonalizing against them somehow?
If the PRNG can take a seed of length log(n) and generate a random string from it, then you might try to take a BBP algorithm and reduce the length of the random input to log(n) using the PRNG. We assume that the PRNG being good enough means that this new algorithm still has the usual BBP guarantee that it gives the correct answer for at least 2/3 of the seeds.
However, now the derandomization of running it on every possible seed just gives a slowdown of 2^log(n) = n, which means that the running time stays polynomial.