To know the meaning of the bear and the star on California's state flag you need to know about the Bear Flag Revolt. That was the first appearance of the Bear Flag.

Thirty-three Americans gathered at the home of General Mariano Vallejo in Sonoma, at dawn, on June 14, 1846. They were revolting against Mexican rule and demanded the surrender of the small fort. They arrested the general and raised a flag over the plaza. William Todd oversaw the making of the flag.
A woman donated a piece of unbleached cotton, and the wife of John Sears tore a red strip of her petticoat and sewed it to the bottom of the flag. Todd drew a star in the upper left corner and drew a grizzly bear next to it. He wrote "California Republic" under the bear.
The flag was raised on June 14, 1846. The original flag was destroyed in 1906, during the San Francisco earthquake and fire.
Here is what happened after the Bear Flag Revolt:
The bear flag was selected as the state flag of California on January 12, 1911. An excerpt from the Senate Bill No. 291 explains specifically what the flag was required to entail:
"The said bear flag shall consist of a flag of a length equal to one and one-half the width thereof; the upper five-sixths of the width thereof to be a white field, and the lower sixth of the width thereof to be a red stripe; there shall appear in the white field in the upper left-hand corner a single red star, and at the bottom of the white field the words 'California Republic,' and in the center of the white field a California grizzly bear upon a grass plat, in the position of walking toward the left of the said field; said bear shall be dark brown in color and in length, equal to one-third of the length of said flag."
Here are some interesting facts about California.