Jared Diamond answers this question beginning at the bottom of p. 94 of Guns, Germs, and Steel. Beginning on that page, Diamond describes a three-step method that is used to determine where a crop or animal was first domesticated.
The first step is to determine whether a plant or animal in the archaeological record was domesticated rather than wild. Diamond says this is done by looking at physical characteristics. On p. 95, he says that “most domesticated plant and animal species differ morphologically from their wild ancestors…” By looking at the remains of the plants or animals, researchers can often determine whether the organisms were domesticated. The second step is to determine the age of the remains. This is typically done by radiocarbon dating. Finally, researchers have to figure out if the plant or animal was domesticated in that place or if it was domesticated somewhere else and then borrowed. There are two ways to do this.
The first way to do this is to look at where the wild ancestors of the domesticated crops lived. You would expect that the crop or animal would first be domesticated in the place where their wild ancestors lived. The second method Diamond describes is to plot on a map the dates of known domestication in each area. If you can see a pattern where later dates radiate out from a central early date, you can assume that the plant or animal was domesticated in that central place.
This is how Diamond describes the process of determining where a plant or animal was domesticated.