READ ME!!!!
My explanation can sound like a distinction without a difference, but it's the impedance match loss that's the key to understanding it.
If the wall is acting like a big speaker cone, like your speaker that's creating the sound, you will have large losses from the impedance mismatch. Speakers are massively inefficient at transferring mechanical energy to the air due to the impedance mismatch between the cone and the air.
Now I should say that some of that IS taking place - it's unavoidable - but the main mechanism is as I describe it in this video.
This is even better illustrated by a thinner material, like a sheet of plastic hung to seal off a doorway between two rooms. Virtually all of the sound passes right through from one room to the other.
BUT, if the transfer method was the one commonly described, you would have huge impedance losses from the sound energy trying to make the plastic move, and then with the plastic trying to move the air on the other side.
Instead the plastic sheet is just moving WITH the sound energy, like it has become part of the air itself. In other words, it's along for the ride.
The sound, in effect, pushes through the plastic as if it doesn't exist at all. Much like my finger presses the button on the control panel on my stove as if the plastic isn't there. My finger presents an overwhelming force to the barrier the plastic provides.
Same with a wall, but the wall (being thicker, stiffer and heavier) is more of a barrier and reflects back more of the sound energy that's acting on it. Only the longer wavelength lower frequencies make it through, and that's why you hear mostly the bass on the other side of the wall.

Barriers do three things with sound: reflect it, absorb it and let it pass through. It depends on the mass, density and stiffness of the barrier how much it does each of these three things.
Thick, heavy, dense walls will let very little sound through and reflect most of it back.
Thin, light barriers let most of the sound through. That's the basis for effective soundproofing.

Acoustic treatment tries to absorb sound. Absorbing the sound converts the mechanical energy to heat to dissipate it faster than it normally would. The goal with acoustic treatment is NOT to soundproof a room, but to reduce the reverb time in the room and cut down on reflections. Acoustic treatment improves the sound quality you'll get in a room, but doesn't do much to stop it from getting out of the room.

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