There's always more to be said on the topic, but I discussed the decipherment of cuneiform in When did the modern understanding of the history of Bronze Age Mesopotamia develop?

In the 19th and 20th century, the field of historical linguistics developed methods for learning how earlier forms of languages were pronounced. These rely on the discovery that changes in sounds are systemic (short "i" might drop out of the middle of words between consonants or every "sh" become "s") and that they occur in some preferred directions (m > n is common, m > k would be very unusual). We can also compare by words written in different languages and writing systems: for example, we might have Sanskrit, Aramaic, and Greek texts that all write the same name or the same word for "horse" and compare if our reconstructions are consistent.

Sumerian is what linguists call an isolate: a language with no known relatives, like Basque. So we can't use these methods to reconstruct it from later languages. But we do have the cuneiform writing system. Each cuneiform sign can be read as a word or grammatical element (logogram) or as a syllable like <ak>, <ke>, <i>, or <bab> (syllabogram). We have word lists written by trainee scribes which have a Sumerian word on one side of a dividing line and a phoenetic (sound) writing on the other side. For example, one might have the sign for "donkey" (ANŠE) on one side of the line and the signs <a-an-še-e> on the other, or "the god Nammu" on one side of the line and <na-am-mu-u> on the other.

And since Sumerian was used to write Akkadian, and Akkadian was a Semitic language with many relatives and many Akkadian words were written in languages like Greek or Hebrew which are still spoken, we can create a fairly faithful reconstruction of the sound values of words by working out the sound values of signs for Akkadian and then piecing them together to understand how the word values were pronounced.

Grammars of Sumerian like Abraham Hendrik Jagersma's tome will talk about the problems of going from reconstructed Greek or Hebrew to reconstructed Akkadian to reconstructed Sumerian. For example, the writing system only has four vowels, and many people suspect that Akkadian had more ... if the writing system could not handle some sounds in Akkadian, what sounds in Sumerian could the scribes not hear and copy as they wrote down how to pronounce the words their teacher recited? But we have a good-enough reconstruction of the sound system in Sumerian that scholars can all write a word or the sign more or less the same way, and get on with arguing about the contents of texts and arguing what those conventional ways of writing Sumerian words in Latin letters actually represent. Its not perfect, but it is better than nothing and lets scholars spiral closer and closer to the truth as they learn new things or work harder on individual problems.

Jagersmaa's PhD thesis is free to download https://openaccess.leidenuniv.nl/handle/1887/16107 and a scholarly physical library will have some less intimidating volumes on paper.