The Rosetta Stone had text in 3 languages, Egyptian Hieroglyphics, Demotic Script (an Egyptian script, what the common people might write in), and ancient Greek.
Ancient Greek was the key to understanding the stone. There were very many scholars of ancient Greek in Europe. Especially since the Renaissance, study in ancient Greek was common. The Collegium Trilingue for instance in Leuven taught courses in Greek, Latin, and Hebrew. Two famous students of this university were Vesalius, the famous anatomist, and Mercator, the cartographer. Humanists were intensely focused on these languages and the texts written in them. Studying the languages wasn't just the endeavors of language, history, or literature scholars, is really was central to many educations.
While the study of Greek was probably not still as prominent by the time the Rosetta stone is being studied, there is still a very good knowledge of the language.
From the Greek they were first able to start working on the Demotic script, and then the hieroglyphics.