I don't understand why they would ever be getting closer to Earth. Wouldn't they be launched at a time of the year when Earth is moving in the direction the they are supposed to go in order to give them maximum speed?
A very simple way to explain it: imagine you are on a playground and there is a merry go round right in front of you. A kid jumps in it and it starts spinning. At the same time you start to slowly walk away in a straight line. If you keep looking at him, you will see that the distance between you two varies (he gets closer, then farther, then closer back again) due to the rotation of the merry go round. In this example, the merry go round is the solar system, the Sun is at its center, fixed in this frame of reference, the kid is the Earth and you are the Voyager probe. Hope it helps!
You cannot really go in straight lines in the solar system due to orbital mechanics. Whatever you do, you're in orbit about something, even if the orbit is a hyperbolic escape trajectory and you're never returning. So the probes were not launched in the same direction they're currently traveling.
I believe the Voyagers didn't gain solar escape velocity at launch; they were sent to Jupiter via a regular Hohmann-like (half-ellipse) transfer orbit and used the gravity assist there to actually kick them to an escape trajectory.
"Because Earth moves around the sun faster than Voyager 2 is traveling from Earth, the distance between Earth and the spacecraft actually decreases at certain times of the year."
The Voyager probes used to be moving much faster, but as they've fought the gravity of the Sun they've slowed down, that's the price they pay for leaving the Solar System. The result is that now the Earth moves faster than the Voyager probes, relative to the Sun.
