The method to prove the Prime Number Theorem can be quickly summarized, and further details can be found in any introductory book on analytic number theory.
The proof relies on understanding the analytic properties of the Riemann zeta function $\zeta(s)$. In particular, the classical approach to proving the Prime Number Theorem is to understand the growth of $\Lambda(n)$, the von Mangoldt function, which is $\log p$ if $n = p^k$ for $p$ a prime and $0$ otherwise.
One can show that $\displaystyle \psi(x) = \sum_{n \leq x} \Lambda(n)$ behaves like $\pi(x) \log x$, and so proving the Prime Number Theorem is akin to proving that $\psi(x) \sim x$. The relation to $\zeta(s)$ is the fact that
$$ -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n \geq 1} \frac{\Lambda(n)}{n^s},$$
so the von Mangoldt function appears as the coefficients of a Dirichlet series whose analytic information is contained in the Riemann zeta function. To get that information, you perform an integral transform whose effect is to pull out the first $x$ coefficients. This is the Mellin transform
$$ \psi(x) = \frac{1}{2\pi i} \int_{2 - i \infty}^{2 + i\infty} \left( - \frac{\zeta'(s)}{\zeta(s)} \right)\frac{x^s}{s} ds,$$
for $x$ non-integers.
Now it boils down to using the residue theorem and understanding convergence. We'll consider the former and waive the latter.
As we move the line of integration to the left, we pick up residues from the zeroes of $\zeta(s)$. The largest zero is at $s = 1$, which has the effect of contributing an $x$. For each additional zero of $\zeta(s)$, which I'll denote by $\rho$, we get additional contribution. In total, we would get
$$ \psi(x) = x - \sum_{\rho} \frac{x^\rho}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log( 1 - x^{-2}).$$
So if there is a zero of real part $\rho$, then there is a contribution to the growth of $\pi(x)\log x$ of order $x^\rho$. The Riemann hypothesis states that the only zero of real greater than $1/2$ is from $1$, which means that the secondary growth terms contribute only things on the order of $x^{1/2}$. In particular, we would have that $\lvert\pi(x)\log x - x\rvert \ll x^{1/2 + \epsilon}$ for any $\epsilon > 0$, and where the $\epsilon$ comes from the fact that we are summing very many zeroes (and the sum contributes something like logarithmic growth).
That's the idea: relate it to analytic information about the zeta function, relate it to the zeroes of the zeta function, handle convergence of everything involved (very technical), and bask in success.
