Another angle I didn't see in the other comments was comparative linguistics. This came later and wasn't used in the initial deciphering, but it's a major part

Another angle I didn't see in the other comments was comparative linguistics. This came later and wasn't used in the initial deciphering, but it's a major part of the modern understanding. Ancient Egyptian was related to other languages, and was part of the rather large still-existing Afroasiatic language family, which includes Semitic (Hebrew and Arabic and others), Berber (of north Africa), and many others. (Importantly, Akkadian is one of the first written languages aside from Ancient Egyptian itself, and it was a very early Semitic language, and not so dissimilar from Ancient Egyptian in some ways.) While only reconstructions, this gives us, in a sense, a model of the ancestral language of Ancient Egyptian. This insight provides more data on possible phonetic values and the nature of that language. There is always a great deal of uncertainty though. By comparison, Sumerian, of similar antiquity, was an isolate and we do not know any language related to it. We have many texts in Sumerian, and many parallel texts between Sumerian and Akkadian, as well as old Indo-European languages like early Persian. We kind of have the sketch of its sounds. But even then, we have less idea what Sumerian sounded like, or how its grammar works, or what its words mean, than we do Ancient Egyptian. And that is despite the fact that Ancient Egyptians didn't write their vowels while the Sumerians did! Having related languages you can compare to is valuable here.