Why does there need to be a copper circle and a wire mesh to detect the waves?
The electromagnetic radio field exerts a force on the charges in the antenna, creating a very small voltage and current measured by the radio. The parts must be good conductors for the charges to be easily moved without much loss.
The copper circle is the basic antenna, and the mesh is a reflector, making the antenna more directional. Were it not directional it would still work, but it would be even more difficult to separate astronomical sources from terrestrial ones.
Why does the copper need to be a circle with a gap?
The gap is the feedpoint, where the radio attaches to measure the charge movement in the antenna. Without a gap, how would the radio attach? There are other possible ways, but a gap works fine and is easy to build.
How would one determine the sizes of the loop and mesh required to measure radio waves with wavelengths that are tens of meters?
With antennas, all dimensions are relative to wavelength. So if you wanted to build a similar antenna for half the frequency or twice the wavelength, double the dimensions. The spacers would become 60cm long, and the loop diameter 106cm.
That said, 21 MHz is a wavelength of approximately 1.4 meters. All the dimensions in this antenna are significantly smaller than that, making it an electrically small antenna, and it may work just as well for a wide range of frequencies with no modification.