The electric poetential in a place is the potential energy per unit charge that would be placed in that position. So, for a particular charge, a difference in electric potential between two places, means a difference in potential energy in those two places.
Now, as everything else in this universe, a charged particle tends to minimize it's potential energy when you lift the restrictions. To make this clear, stand up and take something not to valuable in your hands. Now watch what happens with that object when you remove the restrictions (your hands): it lowers its potential energy as much as possible.
Now back to the conducting spheres. Let's assume they start at a different potential. That means free electrons on one of the spheres can lower their potential energy by moving to the other sphere. However, there are restrictions: the electrons can't travel through the air between the spheres. This restriction is lifted when they are connected with a conducting wire. Free electrons move from one sphere to the other. As this happens, the potential in both spheres change. Free electrons will continue to flow from one to the other as long as the potentials are different, but eventually, they become the same. Now, there is no longer a difference in potential and the charges stop moving. When this equilibrium is reached, and only then, the potentials are the same.