Kidney transplants fail when your immune system attacks the foreign organ. The first successful transplant used identical twins, so no drugs were needed. Later, doctors used radiation and immunosuppressive drugs like azathioprine and prednisone to trick the body into accepting non-identical donor kidneys.
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That is the first paper to ever show that in the literature, but it makes perfect sense. The only time you wouldn't have or mount a rejection to get something that was implanted would be if it was from your genetically identical twin, if you had one. And that's how the first kidney transplant was done. That didn't require any medication because it was from a genetically identical individual. If you are a non identical twin, which was the next step in organ transplantation done by Dr. Joseph Murray, who also ends up was a plastic surgeon that required whole body radiation. And then when they started cadaver transplantation with kidneys, it required azathioprine and prednisone. And that was all meant to help block the patient's immune system response to the transplanted organization.