To give a slightly longer answer, this goes back to the Prime Number Theorem, which asserts that the number of prime numbers less than n (implemented in the Wolfram language as PrimePi[n]) is approximated by the function LogIntegral[n], in the sense that

Limit[PrimePi[n]/(LogIntegral[n]), n -> Infinity] == 1

The nth prime number Prime[n] is the inverse function of PrimePi, and so we can describe its asymptotic behavior by inverting the LogIntegral function. This gives the series in question.

For a really good introduction to the Prime Number Theorem, I would recommend H.M. Edwards' book "Riemann's Zeta Function".