A lot of different ways. One of the most reliable is to compare it to relatives – few languages are completely distinct, as many evolved as tribes grew and split and sometimes there are surviving descendants that can be traced back. Sometimes artifacts like poetry can help, by pointing out for instance rhyme schemes.
If the language is truly dead, they can’t. It is mostly guesswork.
They can get a rough idea sometimes by finding another language that isn’t dead and finding old writings comparing them. They also will sometimes find that different modern languages share words from the dead language and pronounce them the same, then work from those to guess at the pronunciation of others.
Also, with a language like Latin, scholars and the Catholic Church kept using it long after the Roman Empire and they both kept records of scholars discussing the changing pronunciations. This not only helps determine the correct pronunciation, but the pronunciation of words from other languages sometimes was compared to Latin, allowing Latin to tell us how other languages were pronounced.
But in reality, it is almost all just guesses. They do not know.
All writing systems are phonetic. Metrical poetry, homophones, loan words, where rhyme is intended, puns, variations over time (languages tend to change in regular ways) all give clues.
Also, languages tend to employ certain selections of consonants and vowels. How far you get depends in part on the writing system – ancient Egyptian wrote only the consonants, which makes the vowels guesswork, but Sumerian had a fuller writing system.
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