In this post-apocalyptic world, probably brought on by nuclear war, pretty much everyone has been exposed to lots of radiation and it seems many of the War Boys have cancer. Nux is having night sweats and enlarged lymph nodes in his neck, so I'm guessing he has some lymphoma, a type of cancer that involves lymph nodes.
In patients with cancers of white blood cells (leukemias and lymphomas), the bone marrow is overcrowded by defective, cancerous white blood cells, so the bone marrow has neither the resources or space to make red blood cells, which are necessary to transport oxygen. As a result, when the cancer gets bad enough (such as when you don't get treatment), blood transfusions will often be necessary. This way the blood recipient can get the donor's red blood cells to prevent anemia and platelets to prevent bleeding. This is not a cure for the cancer, but it should help with some of the symptoms.
Max probably has blood type of O-, making him a universal donor, so people use him as a "blood bag."