How does inflation drive Ω close to 1? I'll keep it simple. How does inflation drive Ω close to 1?
 A: There are a few different perspectives from which one can answer this question. Here's one.
$|\Omega-1|$ is a measure of the curvature of the Universe. To be specific, it's essentially the ratio of the Hubble length $1/H$ to the curvature radius $R_c$:
$$
|\Omega-1|={H^{-1}\over R_c}.
$$
(To be honest, there may be constants of order 1 in there.) 
The Hubble length is the length scale set by the expansion rate of the Universe, and the curvature radius is the length scale on which the effects of curvature become noticeable.
As the Universe expands, the curvature radius expands with it. A large curvature radius means a Universe that's close to flat: curvature is hard to notice until you go out to large scales. During inflation, the Universe expands rapidly, so $R_c$ grows to be very large. On the other hand, the Hubble length remains roughly constant, so $|\Omega-1|$ plummets.
More geometrically, what we're saying here is that the length scale of curvature (the size of any "wrinkles" in space, if you like) grows, but the natural length scale to use for comparison doesn't. So curvature becomes less and less noticeable.
During "normal" expansion, both length scales grow, and in fact when the expansion is decelerating, $H^{-1}$ grows faster than $R_c$, so curvature becomes more noticeable -- that is, $|\Omega-1|$ grows.
