This is known as the 'Flatness Problem'. Measurements of the cosmic microwave background show that the universe is either flat, or extremely close to it. This means that, if the universe is closed, yet has a negligible curvature (like the earth, you cannot notice the curvature locally.), it must have been even closer to the critical density near the beginning. A slight difference between the critical density, and the density of the universe, would have been dramatically increased over time with the expansion of the universe. So, since the universe is very flat now, it must hve been even closer to the critical density after inflation.

Look on page 9. Equation 11 is the Friedmann equation, basic to all of cosmology.
He manipulates that and immediately gets equation (16).

Then look on page 11 and you see just what you are talking about derived from (16) by easy algebra steps: equations (27 thru 30).

None of that is very "deep". It is just a few algebra steps.

Now the challenge would be for someone to come up with a verbal-intuitive explanation so you feel you understand WHY such a dramatic sounding result comes out of the Friedmann equation (the model of the cosmos that everybody uses. Friedmann derived it from Einstein GR in around 1923 and it's never been improved on. Gives an excellent fit to the data.) Sometimes simple algebra leads to a dramatic sounding result and people want to know why it does. that could be the "few nuggets of wisdom" you are asking for.

I'm not sure I can provide such a nugget. One way to understand would be to look at equation (16) and see that for the U to be near spatially flat Omega (the ratio of actual density to ideal flatness "critical" density) has to be near one.
So the reciprocal of Omega has to be near one.
So that term (Ω^{-1} - 1) is a CLEAR MEASURE OF HOW BAD THE SITUATION IS.
If it starts growing in either pos or neg direction your universe is doomed (to unflatness )

But (16) says that it in fact grows big as densityxR^{2} gets small. R is the scalefactor and the matter density, for example, falls off as 1/R^{3} (bigger volume→lower density). So density x R^{2} falls off as 1/R.

So that measure of unflatness (Ω^{-1} - 1) grows proportionally with R itself, the scalefactor of the universe. (sometimes called "average distance between galaxies" since we don't know the overall size).

In a radiation-dominated stage of development density falls off as 1/R^{4} so
density x R^{2} falls off like 1/R^{2} and that measure of how bad things are getting (if you love flatness) grows as R^{2}. Even worse news than in matter-dominated circumstances.
If the spatial size of the cosmos goes up by a factor of 1000 then the badness goes up by a factor of a million.
So it has to be very small to begin with.

It seems like the key to understanding is to get an intuitive grasp of (16)

But according to the inflation theory our observable universe is only a tiny fraction of the total universe. Won't it thus appear flat now matter the overall shape of the total universe?

The way I usually like to think of the spatial curvature is that it relates the rate of expansion to the density. A spatially-flat universe is one where a particular relationship between the rate of expansion and the density holds. So if our universe had started out with only a very slightly faster expansion, then it would today have a strong negative curvature, and things would have flown away from one another so fast that no galaxies would ever have formed.

Similarly, if the very early universe was expanding only a little bit more slowly, then the expansion wouldn't have been able to overcome the mutual gravity of the matter, and our universe would have recollapsed back on itself in a very short time.

What this means, then, is that something must have happened in the very early universe to cause the rate of expansion and the density to match up to an extreme degree of accuracy. Inflation does this.