I wonder how Hollywood became the place were movies are made.
Hollywood began as a neighborhood in 1870. 
And when movies began to be produced there?
I am not sure the exact date but in 1910 it was consolidated  with LA. 
Any idea what was the first studio to open in Hollywood?
Good question.  I have no idea.  But classic cinema developed from there between 1917 though 1960. 
And are most movies still produced in Hollywood?
I highly doubt it since film production companies are established all over the world. 
What other places are famous for producing movies?
Cinema of the Philippines
The cinema of the Philippines (Filipino: "Pelikulang Pilipino" or "Sine Pilipino") began with the introduction of the first moving pictures to the country on January 1, 1897 at the "Salón de Pertierra" in Manila. The following year, local scenes were shot on film for the first time by a Spaniard, Antonio Ramos, using the Lumiere Cinematograph. While most early filmmakers and producers in the country were mostly wealthy enterprising foreigners and expatriates, on September 12, 1919, "
to Hollywood, Anna, Gene and John moved there with Elias. They arrived in Hollywood in 1937, and moved into a wonderful neighborhood where Bob Hope lived down the street. During this time, there were so many musicians who were out of work that Elias formed what later became the Hollywood Bowl Symphony Orchestra, and began to write music for movies. It was also at this time that he embezzled the entire payroll of the orchestra, and left for Mexico City, on the run, because Mexico did not yet have