Images of Saturn's narrow and contorted F ring returned by the Cassini spacecraft have revealed phenomena not previously detected in any planetary ring system. The perturbing effect of the inner shepherding satellite, Prometheus, seems to introduce channels through the F ring and a 'streamer'-a line of particles that link the ring to the satellite. The detailed mechanism for the formation of these features has been lacking an explanation. Here we show that these phenomena can be understood in terms of a simple gravitational interaction as Prometheus approaches and recedes from the F ring every 14.7 hours. Our numerical models show that as Prometheus recedes from its closest approach to the F ring, it draws out ring material; one orbital period later, this affected region has undergone keplerian shear and is visible as a channel, in excellent agreement with structures seen in the Cassini images. Prometheus' periodic disruption of the F ring will become more pronounced as the two orbits approach their minimum separation in 2009. The model predicts that the appearance of streamers and the associated channels will vary in a regular fashion on a timescale of one orbital period.