A bounce umbrella works very simply by taking the light from a light source and expanding it’s spread.
A bounce umbrella is typically lined on the interior with a silver, gold or white reflective surface and on the outside with black to reduce light from escaping.
To show you exactly I’ve put together some diagrams that show where the light goes. For this example we’ll assume that an umbrella is a cut of a perfect sphere. The focal point, through which all the reflected light passes, is 1/2 * radius of the circle, or half way between the sphere’s center point and the edge of the umbrella.