One of the consequences of flatness is that the dimension of the fibers stays constant. Maybe this is the reason for the term. I'm not sure whether this makes so much sense though. After all, the alps stay three dimensional all the time, but they don't really count as being "flat". But it probably would even much harder to climb them if they had one more dimension...
At least if you think about something that has 0-dimensional fibers all the time and suddenly aquires one two-dimensional fiber flatness really makes sense in the usual sense. I tried drawing a picture here, but mathoverflow always eats my ascii art.