How do sound waves travel through solids?

Sound waves are longitudinal waves, they propagate though space from particles colliding with each other.
Gases are less dense than liquids or solids, so when sound moves through them, the gas molecules bump into each other less frequently because they are more spread out. This causes the velocity of the sound wave to normally be small, audible to our ears.
In liquids, the speed of sound is faster because the molecules are more closely compacted, so the wave can propagate faster. Think of it as like a metal mosh pit. In a heavy metal gig, there are a lot of people (we're going to think of them as particles) closely compacted exerting repulsive forces on each other. When the moshers push against each other, the velocities of the waves of them pushing each other get faster, because they are closer together. If the mosh pit were to spread out, the velocities of the waves would decrease, and the pit would be less intense, like a gas instead of a liquid.
In solids, the molecules are composed in a lattice with a lot of strong intermolecular bonds. This causes the molecules to be really close as solids are very dense, like the mosh pits of the most br00tal bands in existence. Because of this, the waves in a solid travel very very fast, and usually are too fast to be audible, and are considered to be more of pressure waves. Pressure waves are part of the same spectrum as sound waves, but are caused by faster waves. The waves are normally just pressure forces, like pushing an object, your hand pushes some molecules, which push other molecules, which push other molecules... in a wave. So most sound waves in solids are just pressure waves.
We hear the sounds emitted from solid objects because of resonance. Resonance is when an object oscillates at a specific frequency that causes it to increase in amplitude. This is simply moving back and forth, like a heavy metal guitar string! The string oscillates back and forth at its resonant frequency, and pushes against molecules of air, which causes the air to push against other air molecules, creating a sound wave since the air is less dense. The wave travels all the way to your ears, for you to enjoy.