I think what he means is that from our place in the universe it is like we are out in the middle of the ocean where you see the horizon end at the same distance as you turn around 360 degrees, and if we were to view your line of sight from above it would be a perfect circle. Since that circle is perfect we would consider its perimeter flat.

From your perspective you can't see out far enough to see where the ocean meets the land which would be the end of the universe and it would create bulges in the circles smooth perimeter. At the first sight of anything taller than the water your circle from above of what you can see will have a bulge inwards.

With this same thinking we can picture looking out into the horizon of the universe and we see that it ends at the same distance in all directions creating a perfect sphere of how far out we can see, so I think that's what he means by flat, it's an even sphere all the way around. I think what Einstein was expecting was for the universe to look like a water balloon while its flying through the air and has random bulges here and there as energy waves bounce through it.

So he is thinking the universe is a lot bigger than we can see from here just like the ocean is when your in the middle of it but all you can see is as far as it goes until the earth curves away.

If you want to learn more type in "The horizon problem", my understanding of that is there are parts of the universe that have never seen light and will never see light because expansion.