Sound is caused by vibrations moving through a medium.
Solid matter is still a medium and tends to transmit sound faster as the particles are closer together and so hit their neighbors faster and easier.
When sound hits a wall, it beats on it like a drum, transferring the sounds from the air to the wall even as it reflects. Then, the vibrations spread through the wall and reverberate out of the other side. This is one of the reasons an “echo” is quieter. As it hits and reflects, some of the energy is lost to the thing it hit.
The reason it is “muffled” is because the tones that could not transmit through the wall either bounced off or were absorbed by the wall, so you only get the frequencies the wall can transmit and usually a bit of added vibration from the wall itself.
This is why “sound proof” areas are designed with either spiky foam that can break up and cancel out those clean vibrations by disrupting and absorbing then without being able to vibrate itself… Or with vacuums.
Many “sound proof” windows, for instance, are made by evacuating as much of “all” of the air from between the panels of glass as we can. Without matter to move through, the vibrations cannot make it to the other pane of glass and, so, the sound mostly stops at the wall of nothingness between the glass panes. Some might seep out through the floor, cieling, and surrounding walls; but, it removes the window as a sound “weak point” and works in tandem with things like insulation to help keep noise contained.
This “breaking up” of sound waves is also why “popcorn” ceilings are popular and why rugs in a tile room can remove echo. They disrupt and absorb the vibrations and, so, literally remove some of the “sound” from the air, absorbing some of the reflecting sound waves inside of the room.
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