Suppose A emits light to Earth at some position X (That is, the rocket is located at position X). If A is stationary with respect to Earth, then if A emits 1 pulse per minute, position X receives 1 pulse per minute (because the rocket is at X), and Earth will receive 1 pulse per minute.
Now A begins to move away from X. At t = 0, A emits a pulse at position X, but at t=1minute, A emits another pulse as before, but this time A's position is X + h (assuming moving at v= h). Now as this pulse travels towards Earth, it doesn't reach position X until time h/c, where c = speed of light.
So while previously the frequency of light at position X was 1 pulse per minute. The gap between the pulses is now (1 + h/c) minutes.
Similarly, as the pulses reach Earth, they will be separated by (1+h/c) minutes. Thus the frequency of the pulses has appeared to decrease (Doppler broadening).
We can see from above, that this broadening is due to light from each consecutive pulse having to travel a greater distance than the previous pulse when the rocket is moving, thus creating a time delay.