"[using] a disabled form of the virus that causes AIDS"
While true, this is a poor way to describe a lentiviral vector, meant to invoke the idea of using HIV to kill cancer in the minds of readers not familiar with modern molecular biology. HIV is a type of virus called a lentivirus, which itself is a type of retrovirus, which means that it takes the RNA genetic code it has packaged in the virion, chemically transforms it into DNA, and integrates this DNA into the DNA of the infected cell. Lentiviral vectors are designed such that they do this part of the viral life cycle, but are engineered to lack the genes necessary to make more viruses, so the integrated virus is dead on arrival.
In this case, the researchers kept the normal HIV surface receptors so the virus would efficiently target and "infect" T-cells from the patient; normally, lentiviruses are given a generic non-HIV receptor so they can infect any cell type you might be using in your lab experiments. The lentivirus genome contained not the normal viral genes, but a chimeric T-cell receptor designed to stimulate an immune response against CD-19, a surface protein specific to B-cells. Once this chimeric gene is integrated, the T-cells will express it on their cell surface, and stimulate the immune system to target and destroy cells that have CD-19 on them; this kills all the B-cells in the body, both healthy and cancerous. This last point is a problem brought up by TFA, that the patient now essentially has a limited auto-immune disorder as the altered T-cells persist in her body and continue to point them immune system to targeting B-cells, leaving her partially immuno-compromised (which is the funny part about using the "virus that causes AIDS" to do this).