The first mechanic that causes redshift or blueshift is gravity. Importantly, this cannot be fully understood without a fundamental understanding of what light is. Presently, light is thought to be different than matter, a massless component of the universe. This is due to our limited perception. Light is waves of infinitesimal particles, like a wave of particles in the ocean. The mass of those particles determines the wavelength of the light observed. As they are all so small in mass compared to atoms, which we use to observe all things, their mass is extremely difficult to detect and so we have falsely claimed that light is massless. Even E=mc2 simply disproves the theory that light is massless.
Having mass, the difference between wavelengths of light is the mass of the particles of the wave. In all cases, the mass is so small that it remains essentially unobservable, however the variation in mass is shown in its energy.
With this understanding of light, redshift and blueshift are more fundamentally understandable.
When light moves towards a source of gravity, the force of gravity pulls on that light and physically compresses the infinitesimal particles into a smaller volume. This is because the particles do not travel at the limit of motion of matter, c, but rather travel so close to it that it appears to have reached the limit and so we mistakenly call this limit "the speed of light", which is an important nuance to note. Instead, those particles further away are sped up more than those closer because the particles closer to the source of gravity are also even closer to the limit of motion of matter. As a result, the infinitesimal particles are compressed together and must reach a new steady-state and they merge into larger and larger systems, which we then observe as a blueshift.
Inversely, when light moves away from a source of gravity, the force of gravity pulls on that light and physically stretches the particles into a larger volume. This functions similarly to the idea of the "heat death of the universe" where energy disperses over a larger volume and systems ultimately become smaller and smaller until there is nothing (for an observer composed of atoms, at least) observably remaining. Gravitational redshift works in the same way where, due to the stretching into a larger volume, the particles must reach a new steady-state and they ultimately shrink to larger infinitesimal masses, which we then observe as a redshift.
In the same way, motion can cause the stretching or compressing of the volume of the light in what is known as Doppler shift. Due to motion, the same end-result of redshift or blueshift arises as the light is either stretched or compressed, respectively.
The other form of redshift or blueshift does not actually exist and instead is caused by gravitational redshift. So-called "cosmological redshift" is importantly not the result of expansion of space. This is a drastically misinterpreted observation. Cosmological redshift is the result of light from all observed distant galaxies being gravitationally lensed so that it travels in a Figure-8 orbital, which then leads to gravitational redshift per distance. One single object causes this to occur and thus it leads to the pattern of Hubble's "Law". For more information on this, See The Big Bang's Big Assumption.