How a plane of symmetry makes compounds lose their optical activity
That's really simple. Imagine two different vessels which are precise mirror images of each other; so are all the molecules inside. Imagine that you shine polarized light on them. Being the mirror images of each other, they have to rotate light in opposite directions. But as the compound itself is symmetric, they are in fact identical, so they have to rotate light in the same direction. How can these conditions be reconciled? Just like that: don't rotate light at all.

So it goes.