What makes the twin Voyager 1 and 2 spacecraft fly?

They were given enough speed to escape the gravity of the solar system. After their initial boost on rockets, each proceeded on their momentum, until they came into the gravitational influence of the planets they flew by. Those fly-bys gave each Voyager craft a 'gravitational boost', that accelerated them, and projected them to the next planet (or, out of the solar system...)  

Here's a chart showing the speed versus time - it shows how the speed of Voyager 2 was boosted by each planetary fly-by, and also plots the sun's escape velocity over distance: 

Energizer Bunny combined with sails.

Ignore the "science" idiots:  EVERYBODY knows they continue to work so well because they used DURACELL batteries instead of ENERGIZER.  Plus they correctly used long life rechargeable ones.

They use winged wheels and vibrations.

Newton's Laws of Motion state that an object will continue to move at the same velocity unless acted on by a force.  That's what they're doing.  There's no need for a method of propulsion once the initial impulse has accelerated them sufficiently.  We are misled by living on a planet, which is a special case where friction and drag, along with gravity, makes objects behave anomalously.

Plutonium isotope 238. Half life 30,000 or so years. It's on the longest arm and called the thermo unit or something like that and provides electricity. Six pounds of Plutonium isotope 239 made Fat Man blow up wiping out Nagasaki.

They are just drifting through space.  There is nothing to stop them.  No air resistance.

They were initially sent into space with rockets to get them away from Earth's gravity.

Then they used Gravity Assist by flying by Jupiter.  This gravity assist speeded them up and now they are just drifting along.

They received thrust from rockets, now they are simply moving because there is nothing to stop them from moving.  There is no friction in space so no reason for them to not keep going.

Newton's first law.  An object in motion tends to stay in motion unless acted upon by something.  IOW a spacecraft, once moving, in the vacuum of space, will move forever until it hits something.  (Or someone!  But let's not think about that.)

inertia keeps them going, they have a small nuclear reactor for power and warmth