Sketch of Kelvin wave dynamics. Regions of high sea level are regions of high pressure. Currents follow isobars (shown in black).
On the western side of the ocean currents flow towards the equator, taking water away from the poleward side of the high pressure regions and depositing water on the equatorward side. As a result the region of high sea level moves toward the equator.
On the eastern side of the ocean currents flow away from the equator, moving water to the poleward side of the high pressure regions. As a result the region of high sea level moves toward the poles.
At the equator the current moves eastward in both hemispheres, producing eastward propagation of the high pressure region.
Compare this diagram with the behaviour of low pressure regions.