Actually knowledge of that accelerated expansion comes not from the Doppler effect, but from cosmological redshift. It's better if first I introduce them more formally:
The "classic" Doppler effect takes place when the sender and the receiver have non-zero relative radial velocity from each other (that is, they're either separating or coming together), and the wave in question travels at a given speed relative to the medium, which also has its own reference system.
Sound can suffer from the classic Doppler effect. If c is soundspeed in the air, and vs and vr are the components of the velocities parallel to the source -> receiver axis, and as measured from the reference system where the air doesn't move, then the relation between the frequence f0 at which the source emmits (NOT the one at which it travels) and f at which the receiver hears, is:
With light is more complicated, as the "medium", space-time, has no favorite reference system. The final result is that (with no accelerations), if the receiver moves at speed v as measured by the source and at that angle over there:
Now, cosmological redshift... instead of relative velocity between source and receiver, it is caused by the expansion of space. Space increases continuously, at a rate proportional to itself. So while a lightwave is traveling from a very distant galaxy, the distance between its oscillations will keep increasing because the space between them expands. This is only noticeable if the photon has been traveling for a stormton of time, but comparing its effects on galaxies at different distances has let us study the rate at which space expanded at previous points in the past, and we've determined that it's currently doing so acceleratedly.
