There are a few ways. As another answer said, the long-lived ones usually form in supernovae and collisions between neutron stars. Ones like uranium-235 and potassium-40 last long enough that some is left over from Earth’s formation.
Some short-lived isotopes like carbon-14 and beryllium-7 are produced by cosmic radiation.
All the elements from polonium to actinium are part of the decay chains of uranium-235, uranium-238, and thorium-232. In a decay chain, the immediate product of radioactive decay is still radioactive. So there will be a series of decays until it reaches something stable. In the case of uranium and thorium, those are isotopes of lead.
As to how they become other elements, it is due to particles leaving the nucleus, and changing the number of protons and neutrons. Changing the number of protons changes, the element, because the identity of the element is defined by the number of protons.
The two most common ways this happens are alpha decay and beta decay. In alpha decay, an atom emits, a particle consisting of two protons and two neutrons. So, for instance, uranium 238 (92 protons, 146 neutrons) emits an alpha particle to become thorium-234 (90 protons, 144 neutrons).
In beta decay an atomic nucleus emits an electron, turning a neutron into a proton. For example, carbon-14 (6 protons, 8 neutrons) goes through beta decay to become nitrogen-14 (7 protons, 7 neutrons).
Other forms of decay exist, such as positron emission and electron capture. In this instance, the nucleus either shoots out a positron (anti-electron) or captures an electron. This turns a proton into a neutron. For example fluorine-18 (9 protons, 9 neutrons) emits a positron to become oxygen-18 (8 protons, 10 neutrons). This kind of decay is less common in nature. In fact, the example I gave, fluorine-18, does not occur naturally.
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