The question here is: "how can they tell that the increase in CO2 in the atmosphere is due to CO2".
Yes they can measure carbon ratios; no, the data does not necessarily mean what they claim it means. Yes, the changes are consistent with burning fossil fuels, but that does nothing on its own to explain why the equilibrium level of CO2 in the atmosphere is higher today than before.
Let me illustrate this:
Bin(t) - Biological intake of CO2
Bout(t) - Biological emission of CO2
Oin(t) - Inorganic absorption of CO2
Fout(t) - Burning fuel emissions of CO2
CO2 = \int (Bout(t) - Bin(t) + Fout(t) - Oin(t)) dt
All four terms are time varying. Isotope ratios only indicate that Bout/Fout are higher than before. They cannot explain why the CO2 in the atmosphere is rising which necessarily requires understanding the other terms as well.
Don't link to realclimate.org; its heavily partisan and apologetic for even bad science and outlandish claims.