Sometimes Languages survive due to tradition or religion. A good example for this is Latin - a language which mostly suvived due to the fact that it is the roof for a whole family of modern day languages and has preserved a lot of meaning over time, but also due to Christianity, which not only preserved a lot of greek texts as early source material, but kept Latin alive as the language in which they formulated their religious texts. This is my answer to a fairly similar question in regards to Latin's survival, and may add to the answer to your question.

Another example for how language became rediscovered is ancient egyptian. For a very long time the exact meaning of the ancient egyptian written language was very much lost. Some people had a rough idea that the symbols and signs could not just be decoration, there was repetition pointing towards a language and symbolism pointing towards a descriptive nature, but there was little that connected what people still new about ancient egypt due to cultural tradition, and what actually may have been written on the walls and artefacts still present all over this country. The answer to this issue was the Rosetta Stone - a large granite block that had been engraved with not just anciet egyptian, but also in Demotic (also egyptian) and ancient Greek.

It was first discovered by French soldiers during Napoleon's invasion of Egypt in 1799. The officer in command recognized the Greek script and concluded, correctly as it turned out later, that it may have been the same text written in three languages to be readable and understandable for all people present in the time of its creation. Eventually, during the early 19th century, it ended up in England, where it still is, and translation efforts begun. At first they didn't tackle the ancient egyptian script, but attempted to translate the common Demotic speech, using royal names they could understand in Greek to reconstruct phonetic meanings for the original names and words. This was, understandably, a very slow and very experimental process, and didn't quiet work until one particularly inventive scholar involved another still known language, Coptic. This turned out to be the key to read the phonetics, meaning the sounds fitting to the symbols, thus allowing for the identification and revival of how these royal names might have been read and pronounced in Demotic and ancient egypt. They could confirm all three inscriptions were translations of the same text, and what started with the names of Kings and Queens mentioned in the Rosetta Stone, soon turned to a way to identify more and more symbols and signs, and soon it became possible to descibe other texts based on what they discovered here.

Another similar example is Cuneiform - an ancient wedge shaped language that was found on clay tabletts, artefacts, and stone engravings, and was used for a number of languages based around ancient Mesopotamia (modern Iraq area). It was in use for several millenia and was used and adapted during this time for multiple languages, Summerian, Old Persian, and Babylonian, amongst others. Eventually the language was lost, but the fact that so many cultures had used it, had preserved literally thousands of artefacts with all kinds of cultural context, right down to clay tabletts with letters or shopping lists. Fairly similar to ancient Egypt, it was this host of artefacts that was the key to eventually understanding what all of these artefacts may have documented, but with the additional difficulty that different forms of this skript evidently could be allocated to different cultures. In order to eventually translate these different languages, they resorted to the same approaches as with the egyptian scripts by identifying patterns and repititions, symbols and meanings, and names, and went from there.

This video by the Royal Institution Channel on Youtube is a guest lecture by Irving Finkel, the curator of the 130k+ pieces strong cuneiform collection in the British Museum. He explains a lot about what I only scratched at above, and is worth the watching.

At any rate, these are just a few examples of how old languages have been identified and deciphered, but it's also important to know that it's still a ongoing process for many of these examples, and that each new generation of scientists and each new generation of artefacts discovered still adds to what we know.