>  I can't quite visualize how a force that acts tangentially to the path can produce radial deviations.

Consider a satellite orbiting the earth on a circular orbit.
If there would be no air drag, then
the attractive gravitational force ($F_g=-\frac{GMm}{r}$) and
the repulsive centrifugal force ($F_c=mr\dot{\phi}^2$) would perfectly balance.
And hence the radius $r$ stays constant, giving a circular orbit.

Now add some air drag. The drag force is acting horizontally.
That means that the angular speed ($\dot{\phi}$), and hence
the centrifugal force ($F_c=mr\dot{\phi}^2$) will decrease.
This destroys the balance between centrifugal force
and gravitational force, so that there is now a small
net force pointing down towards the earth.