How do waves transfer energy?

Waves can be thought of as propagation of disturbance without movement of matter. We come across waves all the time: light waves, sound waves, radio waves, and so on. With the propagation of disturbance, energy is also transferred. Think of sound waves when someone speaks to us. Air particles get disturbed and that disturbance of the air particles travels towards us enabling us to hear what was said. In this case, each air particle that gets vibrated transfers the energy to the adjoining air particle and comes back to rest, thus propagating the energy through the air. In this motion, the air particles themselves remain (more or less) at the same place, while the motion takes place and energy is transferred.
Similar is the case with water waves—say, in an ocean. Each wave transfers the energy to the next wave, thus enabling energy transfer over large distances without much actual movement of the mass of water.
Another example of energy transfer is the kinetic energy transfer between molecules during an earthquake. When an earthquake takes place, the seismic (or earthquake) waves travel by transferring the kinetic energy from particle to particle. This may result in the momentary motion of the ground until the wave passes.
In summary, one can say that waves transfer energy by disturbance or vibrations of the particles; in other words, particle-to-particle energy transfer takes place.
In case of electromagnetic waves, a medium is not needed for wave propagation. Such waves transfer energy by oscillations of the electric and magnetic fields.
Hope this helps.