Where is the mirror plane in 1-bromo-1-chloroethene?
For all symmetry operations, you should assume the atoms reduced to a point size. Then, if you have a perfectly flat molecule (like in the case of your disubstituted ethene), all atoms will be in the same mathematical plane meaning there is nothing above or below the plane.

Therefore, the plane is an element of symmetry that puts the nothing above the plane on the nothing below the plane and leaves the atoms themselves (in the plane of symmetry) untouched.