If you go to Yemen, or to Yemenite Jewish neighborhoods in Israel, you can still find soft matzah. This is probably similar to what the Jews of the Exodus carried with them as they left, as Exodus 12:34 teaches us that the Jews took their matzahs bound up in their clothes and carried them on their shoulders. Today's cracker-like matzahs would break under those conditions, but not Yemenite matzahs.
What makes matzahs cracker like is the perferations rolled into the dough. This allows air to escape during baking. The earliest opinion I know of concerning this type of cracker-like matzah came from the Rema (Rabbi Moshe Isserles ca 16th century) who held that matzah should be rekikin (defined by some as wafer-like or "thin"). (Rema, Shulchan Aruch 460:4) I understand that a commentator to the Rema, the Baer HeiTeiv (1730 - 1770) noted that soft matzos were available still in his era. It appears from my reading, that the cracker-like matzas were more of a result of mass production and the need to keep thme on the shelf longer. Soft matzas do not have a long shelf life.
Although it appears that many Ashkenazic Jews came to assume that perferating matzah dough was a requirement for it to be considered kosher for Passover, Rav Hershel Schachter, שליט"א, holds that this is not a requirement, and that Ashkenazic Jews can eat Sephardic soft matzah, citing as authority the Mishna B'rurah who speaks of "Matza made as soft as a sponge" which can be used for the Mitzvah of eating Matza.