How do you breed mare in the game championship horse trainer

It's done partly automatically and part on your own. When your horse starts getting better in races and earning money the owner of the stud farm you are at will start bringing mares to you from time to time. You will know this because the owner will come and get you like he does when collecting weekly rent.
Then you will have a choice to either breed (groom) to that mare or not.
If you choose to do so, you will see the horses get transferred into the stable and you will see a bar that says Breeding in the lower right corner. You will have to wait for it to fill up just like you do when your horse is eating or resting. You can also do a time jump on it but I don't recommend that because the game tends to glitch and get the mare stuck in her pregnancy after breeding.

It will take about 3 days for her to give birth after breeding and then another few days for the foal to grow up. When it does you will have an option to either sell it or buy it for yourself.

Hope this helps. :)