Does a battery's voltage fall to 0 V when it has run out?
That looks like an optical wireless mouse. If so it will have an LED to illuminate the surface you're using it on so that the camera (yes there is one - it's only got, maybe, 64 pixels or so) can see the irregularities in the surface. LEDs have a certain minimum operating voltage and this is about 1.8 V to 2.0 V for red, the most common used in these mice.

Figure 1. LEDs require a certain &quot;forward voltage&quot; before they will pass enough current to turn on. here you can see that to get 5 mA or so through a red LED that you'd need &gt; 1.6 V, not including the voltage drop across whatever switches the LED on when you wake the mouse up. Image source: LEDnique.
In addition the rest of the electronics, including the radio transmitter, will require a couple of volts to work.
Engines using petrol from a tank work at full power until the fuel runs out. Batteries are chemical cells and do not perform in the same way. The voltage remains reasonably constant as the chemistry is used up but at some point the voltage starts to fall off quite rapidly.

Figure 2. Various AA cells output voltage with respect to time for a 500 mA discharge. Your mouse is probably using &lt; 15 mA. Image source: PowerStream.
Note that a 1 V reading on a 1.5 V battery does not mean that 2/3 of its capacity is still remaining. It's nearly flat.