Why do batteries recover after a load is removed?

The actual process is dependent on the type of battery we are talking about. In a lead acid battery,

The cell voltage will rise somewhat every time the discharge is stopped. This is due to the diffusion of the acid from the main body of electrolyte into the plates, resulting in an increased concentration in the plates. If the discharge has been continuous, especially if at a high rate, this rise in voltage will bring the cell up to its normal voltage very quickly on account of the more rapid diffusion of acid which will then take place.

from here.

In general, you can think of it as a normalizing of the chemicals involved. There isn't any more "life" in the battery, the life remaining is just in the correct place to give you a little use.

Also, read this answer for basic battery info

Imagine that you have a large metal bar which is heated to some high temperature (say, 1000C), and you dunk one end of the bar in a bucket of cool water. Even if you have left the end of the bar in the water long enough that its temperature fell below 100C (evidenced by the water stopping boiliung), the rest of the bar would still be at a much higher temperature. If you remove the end of the bar that was in the water, it would receive some heat from the rest of the bar, and its temperature would increase. Not to the original 1000C, but to something well over 100C. If the end of the bar was again put in water, more of the water would boil. The longer the end of the bar is left in the water, the cooler the rest of the bar will get. Conversely, the more time the end of the bar is left out of the water, the closer its temperature will become to that of the rest of the bar.

Batteries (and large electrolytic capacitors) exhibit somewhat similar behavior. They can be thought of holding a mixture of current-storing stuff and current-carrying stuff. Only the current stored in the stuff nearest the terminals can be output quickly. Only when the voltage potential in the current-storing stuff nearest the terminals starts to fall can the current-storing stuff further away start supplying current to it; its ability to do so effectively is limited by the amount of current-carying stuff. Given time, all of the current-storing stuff would tend toward the same potential, just as the entire metal bar would tend toward the same temperature, but when a battery is discharged quickly much of the current-holding stuff won't have had a chance to supply its energy.

BTW, in battery construction there is a trade-off between current-holding stuff and current-carrying stuff. A battery which can release 90% of its stored energy in 5 minutes will generally not be able to hold as much energy as a battery of the same size, weight, and chemistry which would take 5 hours to supply 90% of its energy.