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I'm assuming that you mean "how was carbon dating shown to be an accurate method for estimating the age of a sample." I'm also assuming that you know that radiocarbon dating is only one of many methods of radiometric dating—that most rock samples, for instance, aren't dated using carbon at all; if they're datable (which not every rock is), the method would be potassium-argon dating, and/or uranium-lead dating, and/or rubidium-strontium dating, and/or fission-track dating, and/or. In fact, there are quite a number of factors that can throw off the accuracy of a given carbon-14 date.
Libby's updated version of the "Curve of the Knowns", with some added data points, looked like this: You'll note that while the fit is still close, not every data point falls exactly on the curve.
Carbon dating has a certain margin of error, usually depending on the age and material of the sample used.
To check on the method before applying it on various historical and paleontological material, Libby chose material of Egyptian archaeology, under the assumption that no other historical material from over 2,000 years ago is so secure as to its absolute dating.
When objects of the Old Kingdom and Middle Kingdom of Egypt yielded carbon dates that appeared roughly comparable with the historical dates, Libby made his method known.
Most scientists today believe that life has existed on the earth for billions of years.
This belief in long ages for the earth and the existence of life is derived largely from radiometric dating.
With initial large margin of error and anything that did not square with expectation, judged as contaminated, the method appeared to work and was hailed as completely reliablejust as the atomic clock is reliableand this nobody doubted.
But as the method was refined, it started to show rather regular anomalies.Lava (properly called magma before it erupts) fills large underground chambers called magma chambers.Most people are not aware of the many processes that take place in lava before it erupts and as it solidifies, processes that can have a tremendous influence on daughter to parent ratios. What they did was take several Egyptian artifacts, whose ages were already known from historical records and inscriptions, and also samples of wood whose ages could be determined from growth rings.Fortunately, there are also ways to detect these and compensate for them. (If you're curious, here's a start: from the facility at the University of Arizona)Libby's basic method has been refined and extended ever since 1949.To me it has been a real eye opener to see all the processes that are taking place and their potential influence on radiometric dating.