Think electrical potential at a point as how much an imaginary positive charge hates to be there. The higher the potential, the more the imaginary charge wants to escape.
Now, in middle of two equal but opposite charges, an imaginary positive charge both does and doesn't want to be there (because of the negative and positive actual charges, respectively).
Since both charges are equal in magnitude, they fight to a tie. The imaginary charge neither likes nor dislikes the location; therefore, the potential is neither negative nor positive. That is, the electric potential is zero.