They had a patient, BRU, a gay man who had lymphadenopathy.

Lymphocytes from BRU's lymph nodes were cultured. To the culture they added many stimulating agents.

They claimed that after a few weeks in the culture they could find the enzyme reverse transcriptase, RT.

In fact, they did not prove the existence of the enzyme, but of reverse transcriptase activity. That is, the ability of the culture to make DNA from RNA.

They put in a culture a long stretch of synthetic RNA, An, an RNA template. To one end of this they attached a small piece of synthetic DNA, dT15, a primer.

The outcome of this was that a long stretch of DNA was found to have been reverse transcribed from the RNA.

This was interpreted as proof that the culture and thus BRU's lymphocytes were infected with a retrovirus.