One point I am missing in both the question and the other answers is the proper context: this is a proof by induction. That means that we assume the inequality for $n$, which is then renamed $r$ so that we can now focus on proving the same result for $r+1$ (which we can then call our new $n$). Therefore, before and after step 1. we are dealing with inequalities that are known to hold, and we can combine them with other inequalities like $rp^2\geq0$ in the usual fashion, as explained in other answers.
However if this had been a non-induction proof, then $(1+p)^n \geq 1 + np$ would just be our goal, and dropping positive terms from the right hand side would indeed be illegal, as this weakens the goal (we would end up proving less than we claimed). Incidentally, the multiplication by the positive factor $(1+p)$ in step 1. would still be justified (as $a\geq b$ if and only if $ra\geq rb$ when $r>0$), but for the opposite reason: in the actual induction proof we are saying "we assume $(1+p)^r \geq 1 + rp$ and this implies $(1+p)^{r+1} \geq 1 + rp+p+rp^2$" while in a non-induction proof we could validly argue "we need to prove $(1+p)^r \geq 1 + rp$, and this will follow from $(1+p)^{r+1} \geq 1 + rp+p+rp^2$" (but then we would get stuck, and in particularly dropping $rp^2$ would be disallowed).
By the way, the book is really wrong in saying that dropping $rp^2$ strengthens the inequality, it really weakens it, as I said above. But what the author probably wanted to say is that dropping the term only makes the inequality more (easily) true, with a greater distance between the two members.