Back around the turn of the twentieth century, when filmmaking was still in its infancy, there were big studios in France, Germany, the Nordic countries as well as other places. How did Hollywood become the center of the industry?
This is really two questions rolled into one:
There really is no singular answer to that question. One oft-cited reason is that, in the 1910s and 1920s, the Motion Picture Patents Company (Started by Thomas Edison, though really a trust of major movie producers of the early film era) controlled patents on various technologies related to filmmaking. The move out West helped escape patent enforcement, but this narrative doesn't really explain the whole story, especially after many of the MPPC's patents became either unenforceable, void, or expired.
Film-making, especially in the early years of industrial moviemaking, required lots of two things: Light, and space. Early cinema would often use tiny sets placed side by side in a warehouse, where multiple films could be shot at the same time; but as expectations shifted and movies became more involved, the need for bigger studios and particularly for location shooting grew. Demand for movies was constant year round, and outdoor location shooting was both economical and appealing for audiences, but making movies in upstate New York in February is not a fun experience even with today's technology; back in 1915, it was nigh impossible. California's dry Mediterranean climate is sunny year round, which is of course a boon to film-making, both for logistical reasons (You don't have to deal with snow, sleet, muddy roads, or heavy rain) and for cinematographic reasons - the film emulsions of that era needed quite a lot of light, and clear daylight is ideal.
Meanwhile, California and the Los Angeles/San Francisco Bay regions in particular, offered very cheap real estate in close proximity to a city. Out East, and especially in New York City, you had to choose between cheap real estate, and good access to downtown. In an Edwardian Los Angeles that was just starting to sprawl, you didn't have to choose.
The final reason is simply that California has a very diverse set of microclimates and geographies, which allowed for incredibly diverse location shooting. You have beaches, forests, mountains, deserts, prairies, and cities. You even have a bit of the wild west nearby, which is convenient as this shift coincides chronologically with the rise of the western film in American cinema.
This touches on a broader question of why the US became politically and economically hegemonic over the 20th century, but with regards to film, the major reason is this: Internal market. The US was a large country with a large population, relative to other industrialised nations at the time. This large urban population meant a big internal market for film, which in turn meant that US film studios had an easy time of exporting their product overseas - films would pay for themselves in the domestic market, and then, in foreign markets, they would go on to outcompete the local product. American film is not, by any means, universally dominant; it failed to establish a strong hegemony in markets that had strong protectionist safeguards (Eg, France) and in markets that had severe cultural barriers (Eg, India). But otherwise, a combination of this competitive advantage and inertia made Hollywood filmmaking the hegemonic standard against which other film industries are compared.