Why doesn't a nuclide of N-14 simply absorb the neutron created by cosmic rays or solar wind interacting with the atmosphere and become N-15, which is stable?
How or why does the resulting Nitrogen-15 become unstable and spit out a proton, of all things?
Neutrons from cosmic rays/solar winds are not thermal neutrons. Particles of such an origin are often as fast as from particle accelerators, even faster. AFAIK, the record of highest measured energy of a cosmic ray proton is 50 J, what is energy of flying baseball.
Formed nitrogen-15 is excited, ejecting a proton.
$$\require{mhchem}\ce{^{14}N + n(fast) -> ^{15}N^{\ast} -> ^{14}C + p}$$
