It has nothing to do with evolution.
Ah, but it does indeed. It shows that variation does in fact respond to environment. It's only one piece of the puzzle, though. No one claims that the peppered moths, by themselves, prove evolution. There's a lot more to it than that.
Take that variation, for example, and you can link it to speciation - which we can observe today. Heard of 'ring species'? The Larus gulls are several subspecies where variants live in a ring around the Arctic. The Herring Gull in the U.K. can interbreed with the American Herring Gull, and the American can interbreed with the Vega Gull in Russia. And so on, until you come to the Lesser Black-Backed Gull in the Netherlands. It basically can’t breed with the Herring Gull. Hybrids are extremely rare and don't seem to be fertile, like mules.
So, is it a separate species? You could breed it with its relative to the East, and so on. But what if, say, the Vega Gull went extinct? Would you have separate species then?
Now, imagine such variations happening across time instead of (or as well as) space, and you’ve got an idea how species actually do form, instead of the ’saltationist’ strawman that many try to imply.
We have a theory how we think this works but we haven't gotten conclusive evidence.
The thing is, we have so gotten conclusive evidence. Here's one you can partially check on your own body. Lay your fingers on the side of your jaw. Now, trace along the edge up to the very top of the jawbone. Notice how close your fingers are to your ear canal. Inside the inner ear are three bones, the ossicles: malleus, incus, and stapes. They are carefully arranged to transfer sound energy from the eardrum to the cochlea as efficiently as possible. How could such an amazing mechanism arise?
It turns out that a classification of dinosaur called the therapsids had two jaw joints. The therapsids are known (by several independent lines of evidence) to be ancestral to modern mammals... and we have a basically complete fossil record of the gradual transition of one of those jaw joints into the modern bones of the inner ear. Fossils representing over a dozen separate stages have been found. Note that intermediate steps were all advantageous, though not as efficient or optimized. Some transitional forms did help amplify sound energy but didn't work while the animal was chewing. We still have problems with that under some circumstances (try to listen to someone while eating celery) but the separation is far more developed now.
(Note that some have even cited the ossicles as 'irreducibly complex'. The more central figures of the ID 'movement', like Behe and Shermer, haven't done so... but I suspect that's because they know enough of the detailed fossil record to dissuade them.)
Or, my absolute favorite - the twin nested hierarchies! Books used to be copied by scribes, and (despite a lot of care) sometimes typos would be introduced. Later scribes, making copies of copies, would introduce other typos. It's possible to look at the existing copies and put them into a 'family tree'. "These copies have this typo, but not that one; this other group has yet another typo, though three of them have a newer typo as well, not seen elsewhere..." This is not controversial at all when dealing with books, including the Bible.
Now, this process of copy-with-modification naturally produces 'family trees', nested groups. When we look at life, we find such nested groups. No lizards with fur or nipples, no mammals with feathers, etc. Living things (at least, multicellular ones, see below) fit into a grouped hierarchy. This has been solidly recognized for over a thousand years, and systematized for centuries. It was one of the clues that led Darwin to propose evolution. (Little-known fact: Linnaeus, who invented the "kingdom, phyla, genus, species, etc." classification scheme for living things, tried to do the same thing for minerals. But minerals don't form from copy-with-modification, and a 'nested hierarchy' just didn't work and never caught on.)
Today, more than a century later, we find another tree, one Darwin never suspected - that of DNA. This really is a 'text' being copied with rare typos. And, as expected, it also forms a family tree, a nested hierarchy. And, with very very few surprises, it's the same tree that was derived from looking at physical traits.
It didn't have to be that way. Even very critical genes for life - like that of cytochrome C - have a few neutral variations, minor mutations that don't affect its function. (Genetic sequences for cytochrome C differ by up to 60% across species.) Wheat engineered to use the mouse form of cytochrome C grows just fine. But we find a tree of mutations that fits evolution precisely, instead of some other tree. (Imagine if a tree derived from bookbinding technology - "this guy used this kind of glue, but this other bookbinder used a different glue..." - conflicted with a tree that was derived from typos in the text of the books. We'd know at least one tree and maybe both were wrong.)
The details of these trees are very specific and very, very numerous. There are billions of quadrillions of possible trees... and yet the two that we see (DNA and morphology) happen to very precisely match. This is either a staggering coincidence, or a Creator deliberately arranged it in a misleading manner, or... universal common ancestry is actually true.
(Single-celled organisms are much more 'promiscuous' in their reproduction and spread genes willy-nilly without respect for straightforward inheritance. With single-celled creatures, it looks more like a 'web' of life than a 'tree'. But even if the tree of life has tangled roots, it's still very definitely a tree when it comes to multicellular life.)
One final bonus point:
Also, this idea that inorganics singularly sprang into organic life one time a long time ago makes no sense.
Thankfully, there's two problems with this. One, nobody claims that's how abiogenesis happened, so it's just a strawman. Two, nobody anywhere claims it's established yet, and nobody teaches abiogenesis as a fact in school. So complaining about dogmatism on that score is ridiculous.