Great question! We have several sources of evidence that suggest the earth is layered.

One source of information is meteorites that fall to Earth. How do these meteorites from outer space tell us about our own planet? Well, geologist and planetary scientists think that small meteorites represent the same stuff (elements) that our planet is made out of. However, the rocks at the surface of the earth are more enriched in some elements (for example, silicon, oxygen, aluminum, potassium...) and more depleted in others (nickel, iron…) than these meteorites. That tells us that there must be complimentary reservoirs in different layer of the earth. For example, there must be layers with more iron and nickel than in the crust to account for the depletion of these elements in the crust relative to meteorites. In fact, we think that much of the iron and nickel are concentrated in the core of the earth.

Field geology can sometimes tell us about the layer of the earth. There are some places where cross-sections of rock show different layers of the crust and mantle. A good example of these field exposures are ophiolites, which are pieces of oceanic crust that have been thrust up on land. In many ophiolites, we can see the upper layers of the earth from the upper mantle through the crust (check out the wikipedia article on ophiolites):

Another line of evidence for different layers comes from the study of rocks known as xenoliths (these literally means foreign rock) that are brought from very deep in the earth to the surface in volcanoes. These rocks have very different compositions than the rocks on the surface of the earth, so we know that they must have come from a different layer. Some of these rocks contain the mineral diamond, which forms at pressures of ~ 50 kilobars (about 50,000 times the pressure of our atmosphere at sea level; Kennedy & Kennedy, 1976). This pressure corresponds to a depth of ~150 km below the surface of the earth. This is how we know they came from very deep.

Another major (maybe the most important) line of evidence comes from the study of seismic waves released by earthquakes. Seismic wave travel through different materials at different speeds, and some waves don’t even travel through some types of material. In general, seismic waves travel faster through material that is denser. Geophysicists can locate the source of earthquakes and measure how fast it takes a wave to arrive at a seismograph station far away. If the wave arrives faster than expected, that usually means that it sped up by passing through something dense. When waves are collected at a lot of stations, we can determine where the dense layer is. This type of technique allows geophysicists to determine the location of the boundary between the crust and the mantle of the earth, where seismic p-waves increase in velocity from ~6.7 kilometers per second to ~8.1 kilometers per second.

Studying seismic waves also led to the discovery that the outer core of the earth is molten (liquid) because it does not transmit s-waves, which only travel through solid materials.

References
Kennedy, C. S., & Kennedy, G. C. (1976). The equilibrium boundary between graphite and diamond. Journal of Geophysical Research, 81(14), 2467-2470.