This is a somewhat generally accepted theory. (I don't have sources handy, though.)

What makes modern matzah hard is the flour/water ratio. When more water is added, the bread is also softer. The issue with soft matzah is that it spoils quickly, and if you need bread for 7 days you need to make matzah during Passover.

Okay, you say, so make some matzah on Passover, right? But matzah is tricky, because if enough time passes between mixing the water and flour, it is considered leavened. (This is a religious definition, not a physical one.) Merely owning leavened bread is prohibited, so European Jews made dry matzah to prevent unfortunate slip-ups.

This custom started within the last few hundred years (again, I don't have sources, so I'm not sure about exact dates). Yemenite Jews still use soft matzah.

Edit: part of the Passover story mentions Hillel, who would wrap matzah around his Passover offering and bitter herbs. This definitely indicates soft matzah, but is much later than the Exodus itself.