The wave is moving to the right, and the particles inside the wave (supposing that the wave is something like a piece of rope) are also moving up and down (in this way making the peaks and troughs of the wave), and remember that as the wave moves right, the particles also have a component moving to the right.
If we consider only the movement relative to the wave (so we are moving along with the wave) we ignore the right movement and only keep the up and down movement.
