The exact formula for the number of fields is of course $$F(n) = \pi(n) + \pi(n^{1/2}) + \pi(n^{1/3}) + \cdots$$ There are $O(\log n)$ nonzero lower order terms each of which is $O(\sqrt{n})$. So the leading term $\pi(n)$ dominates and the Prime Number Theorem gives the asymptotic $F(n) \sim \mathrm{Li}(n) \sim n/\log(n)$.

As observed by Rob, there is exactly one field for each prime power order. The exact formula for the number of fields is then $$F(n) = \pi(n) + \pi(n^{1/2}) + \pi(n^{1/3}) + \cdots$$ where $\pi(x)$ counts the number of primes up to $x$. There are $O(\log n)$ nonzero lower order terms each of which is $O(\sqrt{n})$. So the leading term $\pi(n)$ dominates and the Prime Number Theorem gives the asymptotic $F(n) \sim \mathrm{Li}(n) \sim n/\log(n)$.