The bristles of a shaving brush are bunched together to ensure that there are a large number of thin columns of air in-between them. When the brush is moved to and fro, the soap solution seeps into the brush, due to surface tension, partially filling the air columns.
While shaving, the soap solution stuck to the skin is dragged to form thin films. But when stretched beyond a certain length, limited by surface tension, the films break and in that process trap a little bit of air to form small bubbles. As there are hundreds of bristles involved in the bubble making process, we get lather within a short time.
The same principle applies to tooth brush and the brushes used for washing.