The best example for this equation is a dragged clump. Imagine a gas on a one-dimensional lattice being dragged to the right by an external force. At each time, each particle is moved to the right by one unit. There are additional random motions of the particles, but they have this drift superposed on top.
If you have a gradient in the particle density, so that the density steadily goes down as you move to the right, if you look at any point, the particles will, on average, come to you, so the density goes up with time. The amount it goes up is proportional to the density gradient (it just comes from the steady motion of the particles).
So the result is that you have a density profile moving to the right at a steady speed. The solution to the equation is f(x-ct), which is a steady clump. If you want a more realistic equation, you can add a $d^2x\over dt^2$ term, which represents diffusion (the diffusion can also absorb a steady drift by a Galilean transformation). When you are only dealing with the first derivative terms, there is no diffusion.