This is a case of the CDC either lying or at least being deliberately misleading with their headline: the study in question shows that amongst people previously infected with COVID those who were also vaccinated were less likely to be hospitalized.

“In today’s MMWR, a study of COVID-19 infections in Kentucky among people who were previously infected with SAR-CoV-2 shows that unvaccinated individuals are more than twice as likely to be reinfected with COVID-19 than those who were fully vaccinated after initially contracting the virus.”