Relativistic wavelength shift for a stationary source and an receding detector is given by
Now, the $\gamma$ can be explained as a result of Lorentz length contraction. In other words, distance between two points in the source frame will be Lorentz contracted when seen by a moving detector.
But how is the $(1-\beta)$ factor explained by the source?
In other words, the source sees the detector detect a higher frequency due to time dilation as well as due to its motion. It also sees the wavelength to be Lorentz contracted in the frame of the detector. How does the source explain the additional factor $(1-\beta)$ needed to explain the same speed of light measured by the detector?