Thinking about the electric circuit model
Adding a second battery to the circuit has the effect of producing a bigger
push from the two batteries acting together, moving the charges around the circuit more quickly. This means that more charges per second pass any point in the circuit and so the size of the electric current is increased. It is worth emphasising here that the number of charges moving around the circuit has neither been reduced nor increased. The same charges simply drift faster because of the extra battery.
There are actually two effects that contribute to the bulb being brighter. The batteries maintain the faster motion of the charges, so that: