At the negative pole, an excess of free electrons is created, and at the positive side a lack of electrons is created (called holes in electronics), leaving just the protons to contribute their positive charge.
The extra electrons in the negative terminal try to push other electrons away or move themselves away and closer to a lone proton in the positive terminal, this due to the properties of charged particles: same charges repel each other and opposites attract.
Electrons can't go to the positive terminal because the negative terminal is isolated from it, or in the case of batteries, there is a chemical reaction keeping them away. All that is left is the force of the electrons pushing against each other, unable to meet a proton and reach an equilibrium.
This contained force is what we call voltage.
When there is an available path through which electrons can move, they quickly reach the holes at the positive terminal; This movement of electrons is called current.
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