If you are needing horses transported from being purchased, they would be under the Horses tab. The drop down menu will have View and In transport. Select In Transport and select Board All At Home. As long as you have enough stalls for it, it will be boarded at your barn, so to speak. 

You can always go in/manage and transfer them to a player run boarding center if need be but I always board them at home and leave them there.

If you do not have room it will say You have boarded 0 horses at home and the horses will remain in the transport area until you can get more room.