A wave transfers energy from one place to another, but it does not transfer matter. The material that the wave travels through only vibrates about a fixed position. It is the wave that travels outwards, not the water or the air.
For ripples on water, drop a small cork onto the surface of a ripple tank. As the ripples pass, the cork moves up and down on the spot. It does not move outwards with the ripples, and after they have passed it is in the same place as before. If the water itself travelled across the tank, the cork would be carried to the edge. The water also does not pile up at the edge of the tank, which shows that it did not travel with the wave.
For sound in air, a lit candle placed in front of a loudspeaker playing a loud note only flickers backwards and forwards. It is not blown out, which shows that the air only vibrates backwards and forwards while the sound wave moves outwards. Similarly, you can hear a person on the other side of a room without any wind or draught reaching you. In both cases the evidence shows that the wave travels and carries energy, but the water or the air does not travel with it.
