How do you spay or neuter your pets? I have seen it but I can't figure out how to do it. What are the advantages?

To make your pets no longer breedable, you'll have to use a philter of infertility or exceptional infertility kit on it, both of which can be bought from the diamond shop. If there are other players selling it, you may be able to find the philter of infertility on the trade broker.

There are no "advantages". Just that some players do not want to sell breedable pets to others, especially if said pets are finished projects after months of hard work and breeding. This prevents others from buying their pets, breeding for free, and producing more similar pets with little effort.

Note that once you make a pet infertile, you cannot reverse it, so the pet will remain unbreedable forever.