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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Because the math works and it has been observed.
When light passes through a gas, some photons of specific wavelengths will be absorbed by the atoms. Each element has specific wavelengths that it absorbs due to electron configurations. The wavelengths that are absorbed will show up as a dark thin line in the spectrum of the observed light. The location and separation of these lines act as a fingerprint for the elements. This is how we can tell what elements make up stars.
When looking at distant stars, astronomers could see these spectral lines in the proper configuration, but they were all shifted towards the longer (redder) wavelengths. In theory these could be brand new elements that just looked like they were shifted, but Occam’s razor (and other evidence) makes this unlikely.
The red shift is why the James Webb Space Telescope is designed to operate in the infrared portion of the spectrum (it is NOT a visible light telescope). The JWST was designed to look at the most distant galaxies visible which are all in the infrared thanks to red shift.
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