People have already correctly noted the extreme emptiness of space and the careful mission planning, but there's another fundamental reason why a probe like Voyager isn't going to start orbiting something (that it isn't already orbiting, like the center of the galaxy).

Newtonian orbits are what's called "conservative". That means they don't gain or lose any energy during the orbit, they just exchange kinetic energy (the energy something moving has) for gravitational potential energy (the extra energy something gets from going uphill against gravity) and back again.

One of the effects this conservation of energy has is that if a probe falls into the gravity well of a celestial body of some sort, the probe will just exchange potential energy for kinetic as it falls into the gravity well, and then exchange kinetic back to potential as it flies away out of it. If it fell towards the celestial body from a very large distance (where the gravity of the celestial body was effectively zero) and had some initial speed, then it wasn't initially gravitationally bound to that celestial object and it won't be gravitationally bound to it in the end, either. It has enough total energy (kinetic plus potential) that it will simply fly away from the celestial body just like it flew towards it and won't go into orbit!

Now, if the probe fires its thrusters, or if it interacts with the planets atmosphere, or if it interacts with a third body, then it is still possible for it to go into orbit of the planet. But the first takes active intervention by the mission controllers, the second requires incredibly precise flying to interact just right with the atmosphere so as not to burn up (and it would come around and interact repeatedly until its orbit did decay), and the third requires an even more improbable and specific three-body interaction where the two-body interaction is already fantastically unlikely.