An electrochemical cell consists of two different conductors (electrodes) immersed in an electrolyte. This setup causes a chemical reaction to occur on the surfaces of the electrodes that generates electrons at high potential energy.
The anode in an electrochemical cell (a battery is a collection of these cells) is the site of a reaction which produces electrons. These electrons move with high potential energy into an external circuit, where they do work on whatever load has been provided. They then arrive at the cathode, where another chemical process consumes them.
An example of the process occurring at the anode might be
While the cathode could involve a process such as
(This was the chemistry of one of the earliest batteries made.)
The negatively-charged anode will be at low electric potential relative to the positively-charged cathode, so the electrons are moving from low potential to high potential, which is the movement that causes them to be able to do work.
Note that a positive charge would have high potential energy when at a location of high potential, but for the negative electrons, potential energy is high when they are at a location of low potential (such as the battery anode).
The amount of potential difference between anode and cathode depends on the particular reaction that takes place in the cell. The reaction produces energy - a certain number of joules of energy for each coulomb of charge released. Since "joules per coulomb" is exactly the meaning of the unit "volt", it follows that the energy of the reaction creates the potential difference between anode and cathode in the cell. This potential difference - the battery voltage - in turn determines the amount of useful work the moving charges can do.