How do we know what the planet is like on the inside?

There are some places where material from the mantle has been brought to the surface, sometimes even in sequence, so we can see the layers. The Kimberlite "pipes" that we get diamonds from also bring up samples of the mantle. We can recognize the different layers by their effects on earthquake waves. For example, we know that there is liquid at the Earth's core because there are particular earthquake waves that will not go through liquids (S waves). Each layer causes "jumps" in the different earthquake waves, so by working backwards from all the different arrivals from earthquakes all over the world, people have determined the layers. The time it takes for the earthquake waves to get to the seismometer tells how fast the material is that makes up the layers, which allows us to make guesses about what those materials are. Also when we find meteorites they may have composition similar to what we find deep in the Earth, so that gives us clues.