The most likely factor responsible for the Spike in Europe´s population between 1800 and 1950 was Demographic Transition. This event is caused by a shift from low levels to high levels of fertility and morality.
This event is usually observed in must modern societies which have high income, and are very well developed. Sometimes, Demographic Transition can also be seen in countries which are still in the developing stages but only partially.
At the same time, Demographic Transition is closely connected to other factors that have a more general process and happen in modern societies. For example, economic growth and modernization or when entire countries raise their level of education, their population tend to increase.
Furthermore, when there is structural transformation from being a farm or agrarian society with basically low production to move to a high productive society, with technological and manufacturing jobs, we also see population increases.
With the industrial revolution many people went from rural areas to urban complexes, and there also was a sharp increase in the population. All these factors are connected with Demographic Transition to make more effective.