Who really funded this research study - "T32-AI-0605"?

The National Institutes of Health (a division of the Dept. of Health and Human Services) is broken up into different Institutes, each of which fund research in that institute's particular area of interest. In the case you cited, CH, SWB, EH, SF, and NB are all principal or co-investigators on grants funded by the National Institute of Allergy & Infectious Diseases (that's what "AI" stands for in the grant numbers), and ST is principal investigator on a grant from the National Institute for Research Resources (RR). As the other poster mentioned, the designation "T32" is code for a training grant, whose purpose is to provide funding of salaries, core equipment and supplies for research training of graduate students and postdoctoral fellows, and "R01" is the designation for an individual, investigator-initiated, research grant.