Hi, MathJakob
When thinking about the surface of the Earth, we intuitively know that it's not flat, because we can imagine it as being a three dimensional sphere. We're 3d beings living in a 3d world, so each of us knows what a sphere is, and we can easily tell if a 2d surface is or isn't flat.
Also, there are three-dimensional effects that give us hints as to the curvature of our planet, like limited horizon, the fact that we can see farther from higher up, or the shape of the shadow of the Earth on the Moon during eclipses.
But imagine if we were two-dimensional beings, with no perception or concept of the third dimension.
In this case, "a sphere", or up and down, would be meaningless as far as everyday intuitions go. We would have to use other means of finding out whether the suface of our world is curved(and how).
The easiest way is to draw triangles and adding the angles. If the geometry is flat, the angles add up to 180°, as in Euclidean geometry. The total being different than that points to a curved geometry of space. If the angles add up to less than that, then the geometry is called "open", if they add up to more than 180°, the geometry is closed.
The effects of the curvature are easiest to notice if the triangles are huge as compared to the total size of the space. A triangle drawn on quarter of the area of a sphere will be hugely deformed(360°!), a tiny one will look almost Euclidean. The smaller they are, the more sensitive the measurements have to be, and as is always the case with measurements, there exists some uncertainty.
The two-dimensional beings on Earth's surface could tell that it is indeed cuved, and closed with a radius of curvature equal to ~6700km, by finding out that the triangles they draw produce angles that add up to more than 180 degrees.
This would allow them to predict e.g. that by going sufficiently far away in one direction, they would come back to the starting point from the other side. Or that two parallel lines would eventually and inevitably cross - all without ever being able to imagine a sphere!
Note that if they were really tiny creatures, they would have to build extremely sensitive instruments, or find a way to draw very large triangles(or both), to reach the same conclusion.
We do exactly the same thing with the Universe, albeit in three spatial dimensions. We cannot look from outside our three dimensions(and in fact there needn't exist any higher dimensions), just as the 2d beings couldn't look outside theirs.
So we measure the angles, by "drawing" the largest triangle we can - by looking at the irregularites in the cosmic microwave background radiation.
We know from theory of the early universe what size the lumpy irregularities ought to be in absolute terms. These lumps provide us with a base of the triangle, with us as the opposing vertex. Depending on the geometry, the lumps would look larger, smaller, or just the predicted size, for closed, open and flat universes respectively.
As far as our observations go, the size of the irregularities appear to be consistent with the predictions of the theories of the early universe, pointing to flat geometry. The uncertainties are such that it is still possible for the universe to be curved, but the radius of the curvature would have to be really huge - over a hundred billion ly, with each subsequent refinement of the data pushing the limit farther away.
The bottom line is: the curvature is best understood in terms of geometry of space. Does it behave as if the space were flat or curved? So far everything looks like it's the former.
