Why must the movement of electrons through the wire be balanced by a movement of ions in the solution?
Karl Ratzch is basically correct, but I will try to provide a little more detail.

Electric charges, when separated, exhibit a very large tendency to try to get back together again.  This is represented in basic Physics by Coulombs Law and the large value of the constant in that relationship.

I presume we are talking about a battery here. If the flow of electrons out of the wire is different from the rate of ions arriving at the plate, the net charge on the plate will change.  Depending on which flow rate is greater, the voltage of the plate will instantly go up or down.  This will then change the rate of both charge flows such that the two almost instantly get back to the same value.

More detail: Assume that the flow of electrons out of the plate through the wire is larger than the flow of anions into the plate from solution.  This reduces the net negative charge on the plate.  This reduces the "force" that is pushing electrons down the wire.  The actual name of that "force" is voltage.  The voltage will change such that it doesn't push electrons out the wire as fast and will repel anions in solution less aggressively allowing more of them to reach the plate (per second).  The opposite is also true.  As a result, the flow in and out of the plate will always remain almost exactly the same, with small differences existing for very brief periods of time when the voltage changes.