Think about what happens when a soundwave "hits" a wall. Really what that means is that there's a high pressure area on one side of the wall (normal pressure on the other) followed by that high pressure area becoming a low pressure area.
So while the sound wave's pressure is high, the air is pushing the wall, causing it to move a bit. This stretches the elastic mediums within the wall (like pushing on a block of jello). Eventually these elastic forces cause the far side of the wall to move, which pushes on the air on the other side, transmitting the sound. When the low pressure region hits, the elastic energy pushes the wall back towards this low pressure area. Once again, this transmits the sound to the other side.
For low frequency sounds this is most of the story. The movement of the wall is relatively fast compared to the period of the sound wave. For high frequency sounds, however, it gets more interesting. With high frequency sounds, the low pressure trough might occur while much of the energy of the sound wave is still propagating through the wall (the jello is still squished, and hasn't had a chance to release outwards towards the other side). Now the elastic energy in this wall is "happy" to go in any direction, so when the low pressure wave starts to form in the air, some of the energy put into the wall in the high pressure phase doesn't ever get through the wall. It is instead "put to work" pulling the wall back to its original shape.
In general, this process has a tendency to invoke the non-ideal properties of the wall. It is not perfectly elastic. Some of that energy gets converted to heat. This "deadens" the sound. No more acoustic energy can be transmitted.
If you look at soundproofing a recording studio, the holy grail is "sprung mass." One common construction is to put up a layer of drywall followed by a layer of elastic and then another layer of drywall which "floats," not screwed into anything. When the soundwave hits this, it has a large mass which takes a great deal of time to accelerate, and a very nice elastic layer to soak up and dissipate the energy. This approach can deaden very low frequency sounds. In normal walls, these elastic effects are typically microscopic material properties which can't soak up as much of the energy into its elastic bonds before transmitting them all the way through.
This is also the source of "resonation." If you time the low pressure wave perfectly, just at the point where the wall is done transmitting energy to the air on the other side, you get to use all of the elastic energy stored up to move the wall, accelerating it a maximum amount. This can actually cause a sound to be louder than it was before, because the wall's movement makes it "easier" on the sounds source to generate higher pressures, providing some of the energy required.