Ik animals in real life get infertile
but how do you get one in game?

To answer the questions in both the title and the body of your post:

"What does it mean when it says 'infertile'?"
This means that the pet can no longer be used for breeding. It cannot be undone. (This is usually done to prevent a player's project baby from being bred to a similar enough pet to create more of that exact design and sell cheap babies out from under the person who spent months breeding their project.)

"How do you get one in game?"
Either infertilize a pet using an infertility philter (1 time use - 100 diamonds) or the Exceptional Infertility Kit (infinite uses - 1,500 diamonds); or buy a pet that has already been infertilized by someone else. It is a deliberate thing that is done to a pet, not something that can just happen randomly. Basically.... think of it as spaying/neutering the pet, not that it was born infertile.

oki..... second question.... How inbred do the pets get?

As inbred as you can make them? If I remember right, the formula that calculates inbred percentage only goes back 10 generations, though. Also inbred percentages do absolutely nothing. They're just for interest-sake. Being inbred does not affect the pet at all.