Most "University of..." schools have "The" at the beginning of their official titles since they are the singular namesake institutions in their respective states. "X State University" names, however, typically do not. Although they are state universities the "X" could be a geographical name, a person's name, or whatever. For instance, you have The University of Mississippi and Mississippi State University.
With OSU, for whatever reason the official name was set down as The Ohio State University, and that wonky singularity has been picked up on as an assertion of elitism, a connotation that's either proud or arrogant, depending on whether you went there.