A brief heuristic sketch of why the PNT should be equivalent to goes something like this:

First note that the PNT is equivalent to x ~ log x.

is a sum of log p values for all p^k < x over all p < x.

There are different primes contributing to this sum.

For each p < x, we can see that log x is very roughly equal to log p^k = klogp, where k is the highest power of p which is less than x. Note that klogp is the total contribution from p and its powers.

Adding all of these approximate total contributions together, we get

log x ~ sum of all log p values for all p^k < x

or

x ~

Complete proof of the equivalence of these statements can be found in A.E. Ingham's The Distribution of Prime Numbers, p.12-13:

Another proof, incorporating other useful relations, can be found in T.M. Apostol's Introduction to Analytic Number Theory, p.74-80.