What's surprising is that in this particular case (the Peano axioms), the reverse is also true; you won't be able to eliminate contradictions by passing to this weaker logic.

Note, all I said is, if you can prove a contradiction classically, then you can prove a contradiction constructively! I didn't say, if you have a classical proof of a contradiction then it's a valid constructive proof of a contradiction. Obviously not! But it will still be true that you'll be able to prove a contradiction constructively; it just won't necessarily be the same proof.

If you want to know more about the particular transformation, well, here's a relevant Wikipedia article: https://en.wikipedia.org/wiki/Double-negation_translation