The Doppler effect is just a consequence of the fact that if an object emitting regular signals (or peaks of a continuous wave) is in motion relative to you, then each signal (or peak) has a different distance to travel to reach your eyes...the relativistic Doppler effect also factors in time dilation, but that's the only difference. For example, suppose a clock is traveling away from me at 0.6c, and it's programmed to send out a flash every 20 seconds in its own rest frame. In my frame, because of time dilation the clock is slowed down by a factor of 1/\sqrt{1 - 0.6^2} = 1.25, so it only flashes every 1.25*20 = 25 seconds in my frame. But that doesn't mean I
see the flashes every 25 seconds, the gap between my seeing flashes is longer since each flash happens at a greater distance. For example, suppose one flash is emitted when the clock is at a distance of 10 light-seconds from me, at time t=50 seconds in my frame.
Because we assume the light travels at c in my frame, if the flash happens 10 light-seconds away the flash will take 10 seconds to reach me, arriving at my eyes at t=60 seconds. Then, 25 seconds after t=50, at t=75, the clock emits another flash. But since it was moving away from me at 0.6c that whole time, it's increased its distance from me by 0.6*25 = 15 light-seconds from the distance it was at the first flash (10 light-seconds away), so it's now at a distance of 10 + 15 = 25 light-seconds from me, so
again assuming the light travels at c in my frame, the light will take 25 seconds to travel from the clock to my eyes, and since this second flash happens at t=75 in my frame, that means I'll see it at t=100 seconds. So, to sum up, the clock flashes every 20 seconds in its own rest frame, and once every 25 seconds in my frame due to time dilation, but I see the first flash at t=60 seconds and the second at t=100 seconds, a separation of 40 seconds. This means the frequency that I see the flashes (1 every 40 seconds) is half that of the frequency the clock emits flashes in its own frame (1 every 20 seconds), which is exactly what you predict from the
relativistic Doppler equation if you plug in v=-0.6c (negative because the clock is moving away from me): \sqrt{\frac{1 - 0.6^2}{1 + 0.6^2}} = \sqrt{0.25} = 0.5. And you can see from the italics above that I specifically assumed the light from each flash traveled at exactly c between the clock and my eyes.
