To understand this, you need to understand how the hieroglyphs were translated in the first place. The Rosetta Stone contains the same text in three scripts: Hieroglyphics, Demotic, and Greek. Some people will tell you that the text is in three languages, this is incorrect. Demotic and Hieroglyphic were two ways of writing the same thing. To draw a comparison, it is basically the same thing as writing "1" or "one". Both are English, both have the same meaning, but each has its own appropriate use.

With the understanding that Demotic and Hieroglyphic are the same language, it is rather easy to pick up the sounds. Especially given that Coptic, which is a language that is still used for religious ceremonies by the Coptic Orthodox Church in Egypt, grew out of (and thus shares many similarities with) the Demotic script. So while our pronunciation may not be exactly right (when is it ever, when it comes to foreign languages? And that's just the living ones), by using linguistic links we have a pretty good grasp of how ancient Egyptian sounded.

For a more in depth look, you might consider James Allen's The Ancient Egyptian Language: An Historical Study or Antonio Lopreno's Ancient Egyptian: A Linguistic Introduction

I don't know much about Egyptian in particular, but I do know how linguists figure this stuff out. This is also more personal knowledge of mine, so I'm sorry to say I don't have sources to provide, so take what I say with a grain of salt.

When it comes to most ancient languages, the comparative method is used. Languages evolve in fairly predictable ways, so often linguists will "work backwards" to get a best guess on the phonology of a language. If you ever see a phonetic transcription with an asterisk before it, such as *f, this means that it is a guess. Egyptian evolved into Coptic, and is in the same family of Arabic, Hebrew, and Phoenician. And sometimes we get lucky with contemporary descriptions of how a language sounds. One such example is a rhyming dictionary for (iirc) Middle Chinese that helped linguists learn with certainty how it sounded.

Another interesting point to note is as a Semitic language, Egyptian used what are called consonantal roots. These are kinda complicated, but basically categories of words are derived from roots consisting of consonants. For a contemporary example, the Arabic root for writing is k-t-b. From this we derive words like kitab, or book, and kataba, he wrote. Due to the predictable nature of consonants in Semitic words, many of their writing systems are abjads, alphabets that drop, or usually drop, vowels. Knd lk hw y cn rd ths (kinda like how you can read this) except with Semetic languages it's even more predictable and easy to read this way. Anyway, all of this means that in some cases we know the root consonants of Egyptian words, but have no way of knowing for certain what the vowels were. One example I know is the god, Ptah, the root of which is ptḥ. While Pitaḥ is a likely pronunciation that we tend to use, we don't know for sure.