why waves transfer energy but not matter
In mechanical waves, each particle (molecule, atom, elementary particle, etc.) will get an energetic push from the nearby particles closer to the source of the wave, then swing (oscillate), and then come back to rest at or near its original position. For each particle, this is like a bouncing ball or a pendulum: the ball or pendulum is displaced (gains potential energy) then bounces or swings (energy switches from potential to kinetic and back again) losing energy with each cycle (mainly through friction) until it comes back to rest.
Because the particles interact with and are bound to each other (e.g., intermolecular bonds), energy is transferred from one particle to another along the direction of propagation of the wave. An analogy would be a string of balls tied together like a string of pearls; when one ball is lifted up, the nearby balls must come along.
In an inelastic (incompressible) medium (like water), the movement will be perpendicular to the direction of propagation of the wave: as the wave moves sideways, the water molecules move up and down. In an elastic medium (like air), the molecules can move back and forth – first in the same direction as the wave and then recovering in the opposite direction, etc. – getting closer together and farther apart, but eventually settling back close to their starting position.
Regarding your last question, the “energy of disturbance” warms up the molecules, but there is no average net displacement once the wave has passed