Nuclear testing in the middle of the twentieth century threw up lots of carbon-14 in the atmosphere. It has been slowly washing out since then. Because we have measured the amount of carbon-14 every year, we can use the amount of carbon-14 in it to simply look up (approximately) what year some plant material grew up in. This version of carbon-14 dating has nothing to do with the nuclear-decay-based carbon-14 dating that most people are familiar with, which acts over thousands of years.

I also found that part weird. I can't find in Wikipedia a clear date. Gemini says that the minimum is "100-300 years" but I followed all the citations but couldn't find the source of that range.

My guess it that they used another isotope with a shorter half life (or is another case of forensic snake oil, that would be a interesting second twist in this case).