How do those even work? I'm very curious.

There are various ways to do breeding projects there are three main categories in breeding projects

Stats: breeding the highest stat of a species by -
1)Leveling up specific breeds and hope for high lvl babies
2) using philers on babies and breedIng them

Color: breeding specific type of pets to have certain colors by -
1) picking / buying certain pets with colors you want
2) dyeing pets the color you want
3) buying essence and choose all the colors u want

Gene/mutation: breeding specific breeds for certain types of genes and mutatins by-
1) picking/ buying certain pets with (new) genes and mutations you want
2) genetic mutate pets (its random and may put on genes you Dislike)
3) buying essence and choosing genes you want

Usually most people do color and gene/ mutation projects together doing projects are time consuming. Try not to b overwhelmed and have fun!

I've seen mostly the color and gene/mutation ones. But the breeding of it confuses me when people generate the breeding pair they want. The genes are all carried most of the time, so how do they get the genes visible?

The gist of it is that two pets with the same carry have the chance to create a new pet with the carry as a visible. E.g. Two Zolnixis with carried tribal runes may have a baby with visible trible runes.
I'm not sure of the percentage chance though. As the same two Zolnixis could also have babys with a carried trait, or no traits at all. ^.^'

A carried gene/mutation could also be passed on to a baby if only one of the parents has it as a carried gene. E.g. Zolnixi 1 has carried tribal runes and Zolnixi 2 has no carries or visibles, the baby has a chance of having tribal runes as a carry.

For projects where people want a six vis pet. They (or at least I ^.^') first try to get a pet with 6 carries so they can progress towards getting a 6 vis pet using the whole carry + carry = visible thing.

Reynbowz and Xavion have made breeding guides that you could look at here and here if you're interested. ^.^

Thank you! That really helps.