How do geologists learn about the Earth's layers without seeing them?

Answer 1

Yes what Howard H has answered, besides drilling, GPR studies and geophysical surveys can give information about the shallow layers. However, the layers within the earth's deep interior are known through wave travel time analysis from earthquake and nuclear detonation data.
thnks

Answer 2

Ah, but they do see the layers of rock, in eroded cuts, in mines, and in man-made cuts.

If you mean the mantle and core, from examining meteors, from the propagation of seismic waves during quakes, and from density calculations.