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 charged particles around the circuit more quickly. This means that more charged particles 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 charged particles moving around the circuit has neither been reduced nor increased. The same charged particles simply drift faster because of the extra battery.
There are two effects that contribute to the bulb being brighter. The batteries maintain the faster motion of the charged particles, so that:
The model to develop is one where adding the extra battery results in more charged particles passing through the bulb per second, with each shifting more energy. So more energy is shifted in each second and the bulb is brighter.