Why is it that as an object moves away from a light source, its light waves will expand and the frequency of the wave will change? As measured in blueshift and redshift, if you were to move closer to an object, its light waves would compress and become "bluer". If you were to move away from an object, the opposite happens. Why is it that the distance between the photons in the wave changes?
 A: First the wavelength or frequency of light has nothing to do with the distance between photons. A single photon has a frequency which is directly proportional to the energy it carries. Second, the red shift or blue shift you are referring to is called Doppler effect and I suggest you have a look on Wikipedia which has an excellent article on the subject: https://en.m.wikipedia.org/wiki/Doppler_effect
This effect is unrelated to the photon nature of light – you can just consider light as a wave to understand this.
A: Everything is relative.
From the point of view of the emitter, it's frequency may be fixed and stable.
From the point of view of an observer, who is moving in relation to the emitter, this may not be true. If the observer is moving toward / away from the emitter at a suitable speed, the observed frequency of the emitter is going to change.
You can experience this with cars passing on the road... it's especially noticeable with horns. As the car passes there is a noticeable shift in pitch, from higher to lower as the car changes from coming towards, to going away.
If you were sitting in the car (or traveling beside it at a more similar speed), the horn would sound like it is at a constant pitch.
This phenomenon is called the Doppler Effect. 
