What is the physical explanation driving current flow in a concentration cell?
Start with a simple thought experiment: pour 100 mL of 1 M nickel (II) sulfate solution into a beaker and very carefully layer 100 mL of 0.01 M nickel (II) sulfate solution on top of the more concentrated layer. Then, even without convection or deliberate mixing, diffusion will, sooner or later, result in the solution having concentration of 0.55 M. In what follows, it is assumed that evaporation is negligible, even on long time scales.
Now consider Fig. 1 below:

In this concentration cell, it is assumed, for simplicity, that the solution volumes are equal in the two electrode solution reservoirs. The salt bridge is 0.55 M nickel (II) sulfate solution, so its concentration is exactly half-way between that of the anode solution and cathode solution.
With the concentrations shown in the figure, the cell potential is +0.02958 V, and the anode is the nickel electrode in the more dilute solution at the left. On the cathode side, the log term is zero, in the Nernst ewuation at upper right in Fig. 1, so the potential is -0.236 V. But on the anode side, the potential is lower by 29.58 mV, due to the log term. So the anode is more negative, the cathode is more positive and electron flow is always from more negative to more positive. Therefore, Ni is spontaneously oxidized at the anode, yielding the necessary electrons, and nickel ions are spontaneously reduced at the cathode.
As a direct consequence of having an external connection, i.e., load or voltmeter, between the electrodes, net oxidation will spontaneously take place at the anode, resulting in an increase in the nickel ion concentration in that solution reservoir. Likewise, nickel ions in cathode solution reservoir will be spontaneously reduced at the nickel cathode and the nickel ion concentration will decrease. The external connection affords the opportunity for something useful to happen, i.e., using the cell as a power source. Otherwise, only diffusion happens (see below).
The cell will be &quot;dead&quot;, i.e., have no more free energy to tap, when all three solutions are 0.55 M. There will be no concentration gradient left.
Now consider Fig. 2:

This is the same as Fig. 1 except that the cell is open circuited, i.e., there is no way for electrons to transfer from one electrode to the other. So all that is going to happen is spontaneous diffusion through the salt bridge. Eventually, there will be just one final concentration at 0.55 M.
During this process, no net oxidation occurs at the left electrode and no net reduction occurs at the right electrode. If a voltmeter is subsequently attached, the reading will be less than +0.02958 V because the concentrations are no longer the original ones: both concentrations are closer, thanks to diffusion, to the 0.55 M center value.
The measured voltage, even using a high impedance voltmeter, is not actually the open circuit voltage. It may be very close, but there is a conceptual issue. In the open circuit situation, no electrons can flow between the electrodes and no net redox processes occur at the electrodes. So the open circuit potential is an ideal potential and the measurement goal is to estimate accurately the open circuit potential while allowing negligible current to flow.
With a high input impedance voltmeter attached to the electrodes, a very small current, i.e., nA or pA, can flow. This only slightly loads down (&quot;perturbs&quot;) the cell potential, so it provides an accurate estimate of the true open circuit potential. And it means that oxidation occurs at the anode, to a very slight extent, and reduction at the cathode, likewise to a very slight extent. Solution concentrations are negligibly perturbed.
So there are two parallel processes taking place. First, diffusion spontaneously acts to equalize the solution concentrations. Second, the concentration cell provides a way of hurrying the equalization process along while extracting useful electrical energy. So the former wastes the available free energy while the latter extracts much of it.
Finally, consider Fig. 3 below:

Now the salt bridge is gone and nothing happens: there is no DC cell potential and the solution concentrations are constant. In figures 2 and 3, there is no net redox happening: for every Ni atom that might get oxidized, there is a nickel ion that gets reduced. Everything is balanced at both electrodes. (So if the electrodes start with highly polished &quot;mirror&quot; finishes, they may get surface roughened by this zero net redox process.)
But once a load or voltmeter is attached to the two electrodes in Fig. 2, this opportunity causes the equilibria to become unbalanced: the solution concentrations must change by increasing (in the anode reservoir) and decreasing (in the cathode reservoir), and that immediately implies that electrons flow from the anode (at the left), through the load or voltmeter, to the cathode (at the right).
Perhaps the following will also help clarify matters. Consider Fig. 4 below:

This shows two beakers, each made of solid nickel, and containing the indicated nickel (II) sulfate solutions. Obviously, this does nothing: it is just two fancy beakers of solutions. Next consider Fig. 5:

Now the two nickel beakers are touching, i.e., in electrical contact. Does this change anything? No. Nothing happens. Next, consider Fig. 6 below:

Now there is simply a wide glass beaker with a solid nickel partition that entirely prevents the two solutions from making contact. Does this change anything? No. Nothing happens. Finally, consider Fig. 7 below:

This differs from the previous figure in having a salt bridge. It is exactly equivalent to using a nickel wire to short the two nickel electrodes in Fig. 2. So the cell is shorted out, there is no cell potential, and the cell will try to supply its short circuit (i.e., maximum) current, limited by the kinetics at the electrodes. So the nickel ion concentration will increase in the anode reservoir and decrease in the cathode reservoir, as expected.