Why do dead batteries appear to recharge on their own after awhile?
Batteries work by way of a redox reaction. The reaction can only occur so fast, thus limiting the current the battery can provide. As more current is drawn from the battery, the voltage decreases.

As the battery becomes depleted, there are less of these reactants available to drive electric current. However, if the battery is left to sit for a while, the reaction can proceed a bit, building up an excess of electrons at one terminal, and a lack at the other.

As soon as the battery terminals are connected with a conductor, a current results as the electric charge imbalance attempts to reach balance. However, pretty soon all that imbalance that was developed as the battery sat on the shelf has been used, and the chemical reaction must continue to create more imbalance to continue driving the current from the battery. If the battery is nearly dead, this reaction can't happen very fast, as nearly all of the reactants in the battery have already reacted.

So, the battery on the shelf isn't "recharging", because it's not gaining any more energy. You are simply giving the chemicals inside the battery more time to react, converting more of the chemical energy that was already there into electrical energy.