It's called immunity. When someone receives a vaccination or survives an active infection with most organisms, they build up antibodies that protect them from a future infection. Transfusions from compatible donors who have survived an Ebola infection have been successfully used to treat some Ebola patients. I believe that is the only "drug" that Dr. Brantley received and he has since donated his blood to help save the lives of others.
Brantly got some ZMapp, but you are right before that all he got was a transfusion from a kid that he had previously treated for Ebola. That probably helped at least until he could get other treatments.