You can think of a chunk of (for example) solid metal as being a bunch of very tiny pieces of mass (atoms) connected together to their nearest neighbors by very stiff and strong springs (the interatomic binding forces).
If you suddenly apply a push to one corner of that chunk by for example striking it with a hammer, two things happen: you set the atoms right next to the hammer in motion, and you compress all those springs in the vicinity of the hammer strike. Those springs then transmit the push from the hammer to the neighboring atoms, which are then set into motion as well.
Those moving atoms then press against all the springs to which they are attached, which then push on the next nearest atoms, and so forth.
In this way, a sound wave gets propagated through the chunk.