How does radioactive decay become radioactive?

That depends on the specific radioisotope. For instance, uranium 238 emits an alpha particle during radioactive decay, reducing the number of protons and neutrons in the nucleus by 2 each and producing thorium 234. On the other hand, carbon 14 emits a beta particle (an electron) during radioactive decay, decreasing the number of neutrons and increasing the number of protons by 1 each and producing nitrogen 14. There are quite a few other examples with different changes depending on the type of radioactive decay.