Unstable carbon-13 isotopes get created at a (roughly) constant rate by the flux of high energy photons in the atmosphere. Plants eat carbon from the air (CO2 molecules) and no new isotopes get created in the carbon that the plant ate. Photons in the air do not go inside organisms, for the most part.
When the unstable isotopes inside the plant (or animal) decay and are not replenished by the atmospheric photon interactions, the fraction of the carbon-13 isotope in the total carbon content of the sample decreases with time. Since the half-life of carbon-13 is known, it is possible to calculate how long ago an organism was consuming atmospheric carbon. (Animals only eat fresh plants so the time delay from plant into animal is negligible.)
Carbon dating is reliable if you assume that carbon-13 has been getting created in the same rate in the atmosphere for a long time. This is a good, and well-supported assumption, but it can only tested indirectly. Therefore, the reliability of carbon dating is only as reliable as the (well-supported) supposition that Carbon-13 production by high energy atmospheric photons has been happening at about the same rate for a long time.
