How and why did the population increase in the industrial revolution?
Prior to the industrial revolution the worldwide population growth was quite stable. It was increasing, yes, but not at the extreme rate seen post industrial revolution.
Many people commonly associate the industrial revolution with machinery and an increase in manufacturing capabilities. Totally true. The rise of machine uses helped create larger cities with increased populations within the cities themselves. That doesn't automatically mean an increase in world population occurred. In fact, the early industrial revolution saw quite a few drops in population numbers across cities. This is because of the lack of sanitation, sewage, and knowledge about diseases and prevention. The industrial revolution did bring much more capable machinery, but also an increase in medical knowledge and research. Louis Pasteur's germ theory arrived in 1865, John Snow published information on Cholera and its prevention, cities implemented public health acts, and even sewage works were put into place. These improvements along with others resulted in fewer deaths. Fewer deaths means a lengthier reproductive life, and more people that are healthy enough to reproduce. That will increase a population.
However, if a population is increasing, but cities can't meet the increased demand of limiting factors such as food, water, and shelter, the population numbers will level off or even fall again. That was not the case during the industrial revolution though, because of the improvements of manufacturing, farming, and transportation. A population could grow large and quickly, because the infrastructure was now able to keep up.