If you want a shortened version the AZT collapses the immune system by inhibiting the formation of ALL immune cells including CD4 cells leading to opportunistic infections and a sevee enough drop in CD4 cells both capable of giving an AIDS diagnosis.
So when someone tests false positive on an HIV antibody test and they are put on AZT or its relatives their immune system collapses devloping symptoms diagnosed as AIDS. Here is more on some of the known causes of false positives: