[mjdn]

Irving Finkel wrote a fascinating book called The Ark Before Noah, which explained how the process worked. Hes also given some great talks about it which are available on YouTube.

There were many breakthrough steps in understanding Sumerian over about 100 years.

From what I remember, (and this is explained better on the wikipedia page for the Sumerian language), in 1838 somebody first decoded the cuneiform script by reading the Behistun inscription of Darius the Great in Western Iran. This was written in Old Persian, Elamite and Akkadian, all in the cuneiform script. The guy who deciphered it could understand modern Persian and so was able to descipher the Old Persian.

From there, others working on a published version of the inscription were able to decipher the Elamite and Akkadian languages. These were all Semitic. Sumerian is non-Semitic in origin, but used a similar writing system. I think the decoding of Sumerian built on this and took place over the next 100 years with many scholars making incremental breakthroughs.