The thing to remember is that the two pi bonds, or the attached substituents on carbons 1 and 3 are not really in a plane as the drawing would suggest. They are actually orthogonal planes to each other because the pi bonds' electron clouds would repel each other so that they're not in the same plane.
Keeping that in mind for the above example, if you say that the CH3 and the H side are on the plane of the paper, the H and the Cl on the other side would be coming in and out of the page respectively. Now if you take the mirror image of that (using your PAT skills), the H and the Cl will be going out and in to the page once you try to superimpose the mirror image on the original molecule.
This wouldn't matter if both substituents on any of the two sides were the same and in that case the mirror image would be superimposable. So for it to be chiral, both sides have to have two different substituents as mentioned above.
Hope that helps