From a right-handed wave, everybody does the first part. Each end dancer is holding right hands with the adjacent center dancer, so they all can turn. This brings the original end dancers into the center, holding left hands with each other. Only these two dancers do the second part. The other dancers are done and simply wait for the center dancers to finish the second turn.
But in a left-handed wave, only the center dancers are holding right hands, so only the centers do the first part. Then everybody can turn by the left for the second part.
Swing Thru can also be done from facing couples. They must first step to a wave, creating a right-handed wave. Then everybody turns by the right, and the new centers turn by the left.