I've explained it before, but I'll do it again.
Imagine you're looking at a Z chromosome. At one point on the chromosome lies the gene that, when mutated from the normal version, gives rise to the Cameo color. At another point on the chromosome lies the gene that, when mutated from the normal version, gives rise to the Purple color.
The first Peach female was bred from a male who was IB split to Cameo and Purple. This means that one of his Z chromosomes had the Purple gene, and the other Z had the Cameo gene.
During the meiosis that leads to sperm formation, the cells divide and replicate their chromosomes, and during this process, chromosomes line up with the other member of their pair, and often wrap over themselves and exchange parts. This phenomenon is called "crossing over" and results in a reshuffling of chromosome parts. Thus the top part of one is exchanged with the top part of another, and you end up with a newly combined chromosome in a sperm or egg cell. Chromosomes passed down to offspring are often a new combination of parts from each member of the pair in the parent.
When this occurred in the male that fathered the first Peach peahen, his two Z chromosomes crossed over, and instead of simply passing down one Z with Purple OR one Z with Cameo, some of his sperm contained a Z with Purple AND Cameo on the same chromosome, and some sperm contained a Z with neither mutation. When a sperm with BOTH mutations on the Z chromosome fertilized an egg, the result was the first Peach peahen. When a sperm with NEITHER mutation on the Z chromosome fertilized an egg, the result was a normal IB hen (see "History of the Peach" on Brad Legg's site, and I also discussed this with the original breeder, Clifton Nicholson).
The only way a male split to Purple and Cameo could have a normal IB daughter is if crossover occurred on his Z chromosomes, giving rise to a sperm that contained a Z chromosome with neither mutation on it. Likewise, it could just as easily occur that a sperm contained a Z chromosome with both mutations on it, and this is what describes the sperm that fertilized the egg which gave rise to the first Peach peahen.
This establishes that Peach is the phenotype which occurs when peafowl have both the Purple and Cameo mutations present on the same chromosome. This means that a Peach male is homozygous for Purple AND Cameo, and if he is bred to a Cameo female, he will have Cameo sons split to Peach, and Peach daughters. This is how it works:
The Peach male has 2 Z chromosomes, each with Purple and Cameo on it. The Cameo female has only one Z chromosome, with Cameo on it, as well as a W chromosome. When they mate, they give one from each pair. The male can give only a Z, and in this case, either of his Zs will carry both Purple and Cameo. The female can give either a Z or a W, and her Z has Cameo on it.
Thus sons receive one Z with Purple and Cameo on it (i.e. "Peach") from Dad, and one Z with Cameo on it from Mom. Since they don't have two copies of Purple, that color won't show. But they do have two copies of Cameo, so it will show. One of their Zs has just Cameo, and the other has Cameo and Purple together (which results in Peach), so they will be Cameo split to Peach.
If one of these males is bred to a normal IB female, half of the daughters will be Cameo, and the other half will be Peach. We'd say he's "split to Peach" rather than "split to Purple" because to be split to Purple, he'd have to have one Z with Purple only -- no Cameo as well, since when they travel together, they're called "Peach."
If we reverse the breeding and do Male Cameo X Female Peach, the sons will be the same -- one Z with just Cameo, and the other with Cameo and Purple. Since they have two copies of Cameo, they will look Cameo. But in this case, the daughters will get only Cameo from their dad, so they will be Cameo.
And for those who still assert that Peach is a separate mutation from Cameo and Purple, here's the test -- if you bred Peach with either Cameo or Purple and all three are REALLY separate genes, then all the male offspring would look IB no matter which way the cross is done. It would be the same as crossing Cameo and Purple -- all sons will look IB but be split to both (as was the father of the first Peach peahen). If you don't get IB-looking sons from Peach X Cameo or Peach X Purple, then the "Peach is a separate mutation" hypothesis is disproven.