A shoot-through umbrella works very simply by taking the light from a light source and expanding it’s spread.
A shoot-through umbrella is typically white and a fairly thin or translucent white at that. The light is intended to travel through the umbrella to reach it’s subject.
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.