I'm not an expert, but here's the story as I understand it. This is
probably close to right.
Start with primary cosmic rays. Mostly protons. Some with ridiculously
high energy. One of them strikes a nucleus in the upper atmosphere
and blasts it to smithereens. The debris includes some loose neutrons.
The neutrons start out with high KE. As we know from reactor physics,
those are not very reactive. When they hit a nucleus, they mostly go
in one side and out the other without reacting.
However, by bouncing around for a while, they gradually lose their KE
and become /thermal/ neutrons.
Thermal neutron meets a ¹⁴N nucleus. It reacts to form ¹⁵N. The
neutron didn't have appreciable KE to begin with, but it picks up a
lot of energy falling into the potential well of the strong nuclear
force. So what you actually get is ¹⁵N*, where the * indicates
"excited state". Generic ¹⁵N is stable, but ¹⁵N* not so much.
It was easy for the neutron to waltz in, because it sees no Coulomb
barrier. On the other side of the same coin, a proton has an easier
time leaving, because it gets a push from the Coulomb potential.
That's in addition to whatever energy it can get from the excitement
of the aforementioned excited state.
According to my limited understanding, this happens fairly quickly and
easily, because it does /not/ involve any weak interactions. Making a
proton via decay of a neutron requires weak interactions and neutrinos
and such, but we don't need to do that. There's plenty of protons in
the nitrogen nucleus, ready to leave as soon as they get a bit of a
push.
At this point we have a loose proton plus an electron that the carbon
doesn't want. Sooner or later that averages out to make hydrogen,
which isn't important to our story.
The rest of the story you know. The radiocarbon reacts with oxygen to
make CO₂. Rain washes it to the ground. Living things take up the
radiocarbon. Dead things don't. So the amount of radiocarbon tells you
how long it's been dead. Calibrate tree rings against radiocarbon
/and/ vice versa to eliminate any uncertainty.