A typical analogy explaining redshift involves a siren or a train whistle. So how do we know the same phenomenon works with light?
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Well, for one, we’ve observed it directly. Redshift isn’t theoretical, it’s a major tool for observing the Universe (both in the form of the Doppler effect for movement and in the form of cosmological redshift for distant objects).
But if you wanted to prove it without observation, you’d go back to the math of relativity. It turns out that for all observers to agree on certain facts, redshift has to occur. For example, suppose you are moving quickly and I am standing still, and a stationary source emits a light signal to both of us. In order for energy to be conserved (which it is, of course), you have to observe a redshifted signal relative to me.
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