Hard work and comparative linguistics, mainly. It helps that while hieroglyphs did fall out of use, the Egyptian language didn't. Since it'll come up in the linked answers, observe the Rosetta Stone; the top and middle texts are both Ancient Egyptian, written in hieroglyphic and demotic, while the BOTTOM TEXT is in Greek.

Of course, more can always be said on the matter, so if anyone would like to contribute further material, don't let this compilation stop you! Ancient Egypt is a most fascinating field of study.

• A sadly now-deleted user has an overview of the 'threads to pull' in reconstructing Ancient Egyptian pronunciation;
• u/ajshell1 has a look as to how it was done;
• and u/XenophonTheAthenian and u/CommodoreCoCo have a few comments on scripts and language here.

Some more material in next post because tag limit.