The fact that adding a second battery in series results in the bulb being brighter makes intuitive good sense to most pupils. Pupils reason that the double source (or battery) is now supplying just one consumer (or bulb) and therefore gives a bigger effect.
A more detailed explanation is, however, more demanding for pupils and is based on two effects which occur simultaneously:
With an extra battery, the positive terminal of the battery becomes more positively charged and the negative plate becomes more negatively charged, and this exerts a bigger force on the charged particles.
As a result:
In other words, adding a battery both increases the current and increases the energy that is shifted by each charge passing.