If anyone else watched that video feeling they understood the "what" but not the "why" like me, let me try to give an explanation. The Riemann Zeta function is fundamentally the link between the counting numbers and the prime numbers. A striking (if unenlightening) showcase of this link is https://en.m.wikipedia.org/wiki/Proof_of_the_Euler_product_f... .

Counting numbers are the building blocks of addition and primes the building blocks of multiplication. The RH is important because most theorems in number theory pertain either to additive concepts or multiplicative concepts, but rarely both. In some sense there is a fundamental link between addition and multiplication that we still don't understand. One can see the collatz conjecture as a byproduct of this fact. A proof of RH would give insight to what this link is and give us a deeper understanding of why prime numbers seem so regular yet random.