Despite having apparently simple explanations, cosmological redshift remains one of the mysteries of the universe. Neither 'expanding space', nor Doppler shift offer a completely satisfactory answer. Here is a better one (perhaps).
What is cosmological redshift?
It is the observed phenomenon that distant galaxies glow at a redder and redder frequencies the farther out they are. Astronomers measure galactic redshift by comparing the absorption lines of certain elements in the light spectrum of the galaxy to the same light spectrum in the laboratory (or to the spectrum of our Sun). The redshift is then defined as the change in wavelength (Δλ) divided by the laboratory wavelength (λ_{0}) of the absorption line in question, i.e.,
z = Δλ/λ_{0} = (λλ_{0})/λ_{0} = λ/λ_{0}  1  (Eq. 1)
where λ is the observed wavelength.
In the expanding cosmic balloon, this is very easily pictured in terms of the ratio of the radius of the balloon now (R_{0}) to its radius (R) at a certain time in the past, i.e.,
z = R_{0}/R  1  (Eq. 2)
as shown in Figure 1 (right). Here R_{0}=100 and the red circles represent earlier values of R. It comes directly from Eqs. 1 and 2. It is not difficult to see why wavelength is inversely proportional to the radius. Or is it?
Figure 1 shows the balloon size from around last scattering of photons (the red dot at the origin, R=R_{0}/1089, representing the CMB) up to today (R=R_{0}=100). The red rings are not time based, but size based, at 25%, 50%, 75% of the radius (or circumference) of today. The picture is valid for any expansion profile, provided that there is at least some expansion (also called a perpetual expansion scenario).
Explanations
In the time that the CMB photons were in flight, the balloon expanded by a factor 1089 and the photon wavelengths were 'stretched' by a factor 1089, giving their redshift as: z = λ/λ_{0}  1 = 1088. Quite reasonable, it seems at first sight,^{[1]} but how can a photon's wavelength be stretched? A single photon does not even have a defined size, so stretching it is not conceptually very palatable!
Another reasonable explanation may be that it is just different frames of reference between transmission and reception of the photon and that the redshift is a coordinate transformation issue, resulting in Doppler shift. However, the two frames of reference may be moving away from each other at greater than the speed of light in vacuum (c), yet we still measure a real, finite redshift. How do we reconcile this fact with the Doppler shift equations of Einstein, which do not work for recession speeds equal to or larger than c?
So, where does cosmological redshift come from?
The balloon analogy offers a neat 'crutch' that makes the phenomenon a little more palatable. It appears as if photons conserve angular momentum as they travel along the surface of the expanding balloon, almost as if they go into a larger orbit around the center of the balloon.^{[2]} Since photons cannot shed speed in order to keep angular momentum constant in the larger 'orbit', they shed linear momentum in another way  by reducing frequency.
The linear momentum of a photon is given by p = h/λ, where h is the Planck constant. Angular momentum magnitude of a photon relative to the balloon center is given by: R p = R h/λ. If R is increasing, λ must be increasing by the same ratio in order to keep angular momentum constant. Increased λ is the same as cosmological redshift.
I suppose in the end it is not too important which 'crutch' you use  stretching of wavelengths, conservation of angular momentum, or even Doppler shift, as long as it is accepted that the received to emitted photon wavelength ratio (λ/λ_{0}) is the same as the expansion ratio (R_{0}/R) since emission.
Jorrie
[1] See the animation on the Webb Space Telescope site.
[2] This is equivalent to Kepler's second law of planetary motion, stating that a planetary orbit sweeps out equal areas around the Sun in equal time intervals. This is the same as the conservation of orbital angular momentum. The same considerations cause any (massive) particle with a velocity relative to the skin of the balloon to conserve angular momentum and hence slow down, provided that the balloon is expanding. The opposite (speed up) happens if the balloon is shrinking. More about that in a followon Blog post.
J

Re: Cosmic Balloon Application II: Redshift