Photon red shifting and Doppler effect

[Sefer Tov]

When thinking of sound waves (or more commonly seen waves) the doppler effect is fairly obvious.
A car is moving away from you, it's horn's membrane is fluctuating, increasing & reducing air pressure over time. So if it's moving away from you the sound's peak would occur further away, effectively lengthening the wavelength.
In this scenario, the frequency is not carried by any specific particle. If I examine any single air molecule, it doesn't contain any information about the wavelength, at best it can indicate (based on its energy) some intermediate phase of the wave.

However, with light, every single photon when created, already has a given frequency. It's frequency does not relate on the existence of any other photons around it, it's a given property of every photon.

In that scenario, I'm unclear as to how the doppler effect affects the photon. Why should it red shift? Meaning why its frequency should change just because the emitting object was moving away from me.
And please, don't just wave at me "that's a direct result from the equations of relativity" since that does not provide a simple explanation (for a layman such as myself) of the mechanics involved.

Also, if the same light source emits light equally in all directions (same energy for all photons) and there's an observer at front of it and behind it - both would receive light in different frequencies (red shifted and blue shifted) which means the photons have gained and lost energy depending on the direction of the moving light source.
I was under the impression that the moving (light emitting) object does not contribute energy to the photon if their vectors of motions have some correlation. (as opposed to classical physics where a man on a train throws a ball, then the ball would have a combined energy of the train and the speed provided by the thrower)

I'd appreciate some insight and explanation about this.

 

[Orodruin]

However, with light, every single photon when created, already has a given frequency.

No it does not. The energy - and thus frequency - of the photon is a frame dependent quantity. Just like the energy of any other particle.

which means the photons have gained and lost energy depending on the direction of the moving light source.

again, energy is frame dependent. This is nothing particular for light. It would be the same for balls thrown from a car.

 

[BiGyElLoWhAt]

Well, you see, if you take the covariant derivative of the inverse tensor along an affine connection...

Have you tried thinking about a photon as an excitation of an electromagnetic field? So the field is oscillating. If you look over a span of time, a wave packet comes in and the first peak hits an object that it's about to rebound off of. By the time the second peak comes into hit, the object has moved away, and thus the second peak actually has a larger distance to travel than the first one, and the third even longer.

This is, of course, a loose explanation.