How do we know what is inside the Earth?

Almost entirely from earthquakes. Seismic waves travel through the planet and arrive at stations around the world subtly changed by everything they passed through, which makes the Earth its own scanner.

The two wave types that do the work:

P waves are compressional, like sound, and travel through solids and liquids.

S waves are shear waves, which require a material that resists twisting, so they travel through solids only.

The decisive observation. After a large earthquake, S waves are detected across much of the planet but never on the far side — a shadow zone covering more than a third of the globe. The only explanation is a liquid layer blocking them. That is the outer core, and it was deduced in 1926 from seismic records alone, decades before anything could be sampled.

How the rest was mapped:

Refraction. Waves bend where material properties change, so abrupt changes in arrival times reveal boundaries. The crust-mantle boundary — the Mohorovičić discontinuity — was found this way in 1909.

Inge Lehmann's 1936 discovery that faint P waves appeared inside the shadow zone, which required a solid inner core reflecting them.

Free oscillations. Great earthquakes make the whole planet ring like a bell for days, and the frequencies constrain density throughout.

The independent checks that matter:

Mass and moment of inertia, from gravity and rotation, prove the Earth is far denser at the centre than any surface rock.

Meteorites, particularly iron ones, are thought to be fragments of similar bodies, supporting an iron-nickel core.

Laboratory work compressing minerals to core pressures in diamond anvils, matching their behaviour to seismic observations.

Magnetism, which requires a convecting, electrically conducting liquid — exactly what a molten iron outer core provides.

Deep drilling contributes almost nothing. The deepest borehole ever made, the Kola Superdeep, reached about 12km — a fraction of a percent of the way, and not even through the crust.