We cannot know with absolute certainty why this happened. We can only make informed guesses. The best guess that we have at this point is that plant and animal domestication arose in the Fertile Crescent because of geographical factors. There are two main geographical factors that are responsible for this.
First, there is the fact that the Fertile Crescent had an ideal climate for growing crops. This area had what is called a Mediterranean climate, with hot, dry summers and winters that are mild and have a good deal of rainfall. This is a very good climate for growing crops.
Second, the Fertile Crescent was home to a large number of species that could be domesticated. Not all plants and animals are that good for domestication, but the Fertile Crescent was lucky enough to have many that were. It was home, for example, to many kinds of grains with relatively large seeds, such as the ancestors of wheat and barley. There were also relatively many animal species that could be domesticated.
Agriculture, then, arose in the Fertile Crescent because it was geographically “lucky” in terms of its climate and the availability of species that could be domesticated.