Dish-shaped reflectors are used to steer microwaves in order to establish communications links between nearby buildings. Those reflectors are often made from metal mesh. Why don't they have to be made from solid metal sheets

A "screen" or even just a set of parallel bars are highly reflective to electromagnetic waves as long as the open spaces are small compared to the wavelengths.

"Grid" dishes work fine ... with less weight and less wind resistance ... for frequencies below about 3 GHz. (Wavelengths of at least 10 cm.)

(I even worked on a microwave system in South America where huge grid dishes were used on a 90-mile link.)

an enlarged virtual image