Why battery units are given in ampere hours?

A batteries voltage changes over its lifetime. The current is set by the circuit it is connected to.

As the current is a constant known value and can be predicted, and the voltage cannot, the units are in the value that can be predicted.

Your electricity supply is a constant voltage and can be predicted.

This will tell you at how much current your battery will drain fully. that means it gives you battery capacity.

An ampere hour or amp hour (symbol Ah; also denoted A. h or A h) is a unit of electric charge, having dimensions of electric current times time, equal to the charge transferred by a steady current of one ampere flowing for one hour, or 3600 coulombs.

Battery capacity could be measured in terms of total number of electrons, but this would be a huge number. We could use the unit of the coulomb (equal to 6.25 x 1018 electrons, or 6,250,000,000,000,000,000 electrons) to make the quantities more practical to work with, but instead a new unit, the amp-hour, was made for this purpose. Since 1 amp is actually a flow rate of 1 coulomb of electrons per second, and there are 3600 seconds in an hour, we can state a direct proportion between coulombs and amp-hours: 1 amp-hour = 3600 coulombs. 

In an ideal battery, the relationship between continuous current and discharge time is stable and absolute, but real batteries don’t behave exactly as this simple linear formula would indicate. Therefore, when amp-hour capacity is given for a battery, it is specified at either a given current, given time, or assumed to be rated for a time period of 8 hours

A power source is specified in terms of its energy capacity . and energy is can be expressed as the product of  volts X Current XTime , Here in the case of battery the voltage for a battery is constant and the rest of the part current and time can be different for different operating condition but the product of the same will be proportional to the total energy that can be delivered by the battery at that specific voltage  . Hence it is specified as Ampere Hours @ the specific Volt as it refers to the total energy that can be delivered by the battery

It is because a battery can supply x ampere for y hours. It means 10AH battery can supply 10 A for 1 hour, or 1 A for 10 hours

Because the number of electrons generated depend on the number and speed of chemical reactions occuring in the battery, the more reactions occuring in unit of time, the higher is the current generated. The simplest expression is to tell the current generated in 1 hour.

How much time the battery runs in hours