I'll take a stab at that, although I'll defer to "Physicist" if I don't get this quite right.....
Inflation is constrained by Conservation of Energy; as space inflates, the volume of space permeated by gravitational fields increases. Since gravitational fields have a NEGATIVE energy associated with them (ask "Physicist" for his delightful explanation of this fact), the amount of negative energy in the Universe increases as space is "inflated." Conservation of Energy REQUIRES the production of EXACTLY the amount of positive matter/energy needed to offset the increase in negative energy, such that the net total is always zero (otherwise, it would violate the Conservation of Energy.)
But that's exactly what flat space is: the gravitational fields are counterbalanced by precisesly the amount of matter/positive energy needed to cancel out the negative gravitational energy. Thus, the Inflationary process has no choice but to yield a Universe that is spatially flat, because Conservation of Energy won't permit anything BUT that from happening.
Just for fun, I'll go one step further: The Law of Conservation of Energy is a consequence of the temporal homogeneity of the Universe (physical properties are invariant under temporal transformation.) So, if Inflationary Cosmology is correct, then it follows that the flatness of space is a consequence of the temporal homogeneity of the Universe...... I think.