You're right to question that claim. Edward Waters University, founded in 1866, is demonstrably older than Rollins College, founded in 1885. The website claiming Rollins is the oldest is incorrect. There's a common error in these sorts of historical claims: sometimes people confuse "oldest continuously operating college" with simply "oldest college." Even then, the claim for Rollins might be debatable depending on how you define "college" and account for periods of closure or significant restructuring at other institutions.
In short, the website is simply wrong. Edward Waters University holds the title of Florida's oldest college.