Measuring the curvature of the Universe is doable because of ability to see great distances with our new technology. On the Earth, it is difficult to see that we live on a sphere. One stands on a tall mountain, but the world still looks flat. One can see a ship come over the horizon, but that was thought to be atmospheric refraction for a long time.
Our current technology allows us to see over 80% of the size of the Universe, sufficient to measure curvature. Any method to measure distance and curvature requires a standard ‘yardstick’, some physical characteristic that is identifiable at great distances and does not change with lookback time.
The three primary methods to measure curvature are luminosity, scale length and number. Luminosity requires an observer to find some standard ‘candle’, such as the brightest quasars, and follow them out to high redshifts. Scale length requires that some standard size be used, such as the size of the largest galaxies. Lastly, number counts are used where one counts the number of galaxies in a box as a function of distance.
To date all these methods have been inconclusive because the brightest, size and number of galaxies changes with time in a ways that we have not figured out. So far, the measurements are consistent with a flat Universe, which is popular for aesthetic reasons.