I skimmed the article, but the next third of the article seems to be devoted to using this relationship between the zeta function and prime numbers to prove the prime number theorem, which is a theorem approximating how many primes are less than or equal to any given number N.
The final third goes into how to get increasingly accurate approximations for the number of primes less than N, ending on the fact that Gausses approximation is in some sense the “best”, but only if the Riemann zeta functions zeroes lie on the critical section.
If you just want a general primer on why the zeta function has anything to do with primes, the product formula might suffice. In which case, the proof on the Wikipedia page might be a better read. The derivation in the article focuses on the general setup that is later built on to prove additional things
Highly simplified, and also wrong, you don't get the actual primes, but an error correcting term to another prime estimation function.
So in a way, the zeroes of the zeta function encode "the frequencies of the primes" (in the Fourier sense)
[1] https://en.wikipedia.org/wiki/Von_Mangoldt_function
[2] Prime numbers and the Riemann hypothesis. Mazur, Stein