This would typically be said about a film, or perhaps a novel. The person who drives the plot in a film is the central character, (or "hero", or "protagonist"), whose actions and decisions shape the events of the film. This is highlighting a change from mainstream commercial films of the past - and the very recent past - when the central characters were almost always male, and the female characters were there as eye candy or love interest - not driving the plot, but accesories to the men who were doing that.

A plot is the main events of a play, novel, movie, or similar work, devised and presented by the writer as an interrelated sequence. To drive something is to take control of it. Thus, in this context they are moving the plot forward; or progressing the plot. The plot, in this case, is unclear. I would assume it is either their life or their political agenda.