Why it's not explained just by Doppler redshift caused by faster movement of those galaxies billions of years ago when that light was emitted?
Would the speeds of the galaxies necessary for Doppler redshift to explain all of the observed galaxies redshift be unreasonable or is there something else that prevents such explanation to be sufficient?
If space were non-expanding the redshifts could indeed just be described as a (relativistic) red shift, but ...
... if space were non-expanding the universe would have to be a very strange place. For a start General Relativity would be disproved (because GR predicts space must be expanding or contracting) but GR passes lots of experimental tests so we're fairly confident it's correct. You'd also have to explain how the Cosmic Microwave Background exists if there wasn't a Big Bang. Finally of course you'd have to explain how come Hubble's Law holds.
So the reason why we say the redshift is due to the expansion of space is because there is so much evidence that the universe is expanding, and the expansion naturally explains the red shifts.
You might be interested in skimming the Expanding Confusion paper by Davis and Lineweaver, which I've referenced many times on this site. In particular, the last sentence of the abstract states:
We analyze apparent magnitudes of supernovae and observationally rule out the special relativistic Doppler interpretation of cosmological redshifts at a confidence level of 23$\sigma$.
The basic idea is looking at the luminosity distance using Type Ia supernovae (the "standard candles" of cosmology). The results you'd predict from a Doppler shift - even a special relativistic one - do not agree with reality. You'd have to have some pretty ad hoc additions to your special relativistic cosmology to get these results. Section 4.2 of the paper covers this.
It's not surprising, though, that there is so much confusion. The authors themselves got confused, and an earlier version of Section 4.1 stated that you could differentiate between Doppler shifts and GR based on timing with standard clocks, but in fact the two effects are indistinguishable. Perhaps the best part of the paper is Appendix B, which gives some 25 examples of misleading or outright erroneous statements on this and related topics made by well-known physicists.
I also should add that we also have the CMB radiation to observe, and it provides a unique local at-rest reference frame. That is, if you are moving with respect to the CMB, you'll know because it will appear hotter in the forward direction and cooler in the opposite direction. Now we have a small velocity relative to it, but nothing close to the speed of light. So if you try to explain redshifts with just SR and the Doppler effect, you have to explain why we happen to live in one of the 100 or so galaxies barely moving with respect to the CMB, and why the billions of other galaxies we observe are all moving quite fast with respect to it. In GR, everyone can be locally "at rest" and see an isotropic CMB, while still seeing each other redshifted. So there's a Copernican argument for you - the SR description forces you to believe you are in a special location in the universe.
If you want to intuitively understand the cosmological redshift, I'll shamelessly plug my own excessively verbose answer to a related question.
Question3 : Why do objects that are farther from the center of the universe move faster away from the center?
My opinion : After an accelerating expansion (such as inflation) of early universe has almost finished, particles started to have some velocity.
This velocity distribution naturally has higher velocity when it is further away from the center of the universe and has lower velocity when it is closer to the center.
A. Big bang simulation in the zero energy universe
[Video for Big bang Simulation]
http://www.youtube.com/watch?v=SRUqQM2FfNU
Fig.1.Velocity distribution of galaxies at early universe. Red arrows show the velocity vector of particles. It can be known that the magnitude of velocity vector is bigger as it become further from the center.
Even if the velocity of particles is zero in the early universe, there are particles with higher velocity in further areas from the center and particles close to the center have relatively low velocity by inflation (an accelerating expansion). When positive mass gravitationally contracts to form a galaxy, momentum must be conserved, so higher initial velocity continues to exist as it becomes further away from the center of universe.
B. Natural distribution of velocity in the 3D space Thinking in another way, 3 dimensional space can be divided into 3 areas (from the center) to far, middle, and close area. Even if the velocity of the far area is lower than the middle area, middle area particles exceed far area particles when time passes because the velocity of middle particles are higher. As a result, velocity distribution of particles shows that the velocity of far areas is highest, middle area is second, and the close area becomes third.
C. Velocity distribution when some kind of anti-gravitational source exists If some kind of anti-gravitational source in 3 dimension exists, M exists with even density, the above velocity distribution can exist.
If anti-gravitational source is evenly distributed in accelerating expansion time like the inflation of early universe, a bigger acceleration a exists as r becomes larger and velocity distribution has a higher velocity as the radius of the universe becomes larger. As a result, higher velocity exists for particles of far area from the center of the universe after inflation ends.
The 3 explanations shown above mean that higher velocity for larger R(distance from the center of universe) after inflation in the early universe isn't a peculiar phenomenon. If speed in small area in the early universe distributes from 0 to c and if some time passes, velocity distribution will be in order as above.
Expansion of the universe and expansion of the space are not the same concept. I agree that universe is expanding. But, expansion of space should be reviewed.
