It might be worth taking a look to the original text, Galileo's Discourses on Two New Sciences. The reasoning you're looking for is on the Third Day, a translation of which may be found online. The relevant parts are labelled Theorem I and Theorem II in the above-linked translation.
To derive that distance in a uniformly accelerated motion (e.g. free fall) goes as time squared, Galileo first argues that
The time in which any space is traversed by a body starting from rest
and uniformly accelerated is equal to the time in which that same
space would be traversed by the same body moving at a uniform speed
whose value is the mean of the highest speed and the speed just before
This is argued on a graphical basis (see above link).
However, even though the pictures may look pretty similar to modern functional representations (e.g. of velocity vs. time) and arguments involve finding equal areas in different situations, the arguments never involve an actual calculation of an "area" with mixed units, which wasn't yet conceivable at the time (e.g. $m/s \cdot s = m$). In fact, the whole Third Day seems very convoluted precisely because the notion of velocity wasn't clearly numerical yet, since only commensurable (same unit) quantities could conceivably be operated with (added, divided,...). Proportions of non-commensurable quantities, however, could be compared (today we'd say they are dimensionless), as in
If a moving object traverses two distances in equal intervals of time, these distances will bear to each other the same ratio as the speeds
(earlier in the Third Day)