Let $f$ be a primitive integral polynomial of degree $d$. In Arithmetical properties of polynomials, Ërdos says (page 417) that the integers $n$ for which $f(n)$ is $d$-th power free have positive density,
by a simple application of the Sieve of Erathostenes and an easy limiting process.
(He makes a similar remark in page 418).
I have nearly zero knowledge of analytic number theory. I have looked at two previous papers on a similar subject, one by Ricci and another by Nagell, which may apply this technique, but I couldn't find it (Ricci's paper seems strangely organized to me, while I do not understand Nagell's because I cannot read German).
Can you explain what Ërdos meant, with some detail? Can you give some guiding examples to learn the technique?