The electrode potential depends on the equilibrium (for example)
M(s) <==> M2+(aq) + 2e
The further the equilibrium is to the right the more negative the electrode potential.
By the equilibrium law:
Kc = [M2+]
M(s) cannot appear in the equilibrium law equation as it's not in the same state.
Therefore an increase in concentration must cause the equilibrium to move to the left hand side making the electrode potential more positive. And a decrease in concentration must have the opposite effect i.e. make it more negative.
This is in line with Le Chatelier's principle.