How and why did Buddhism spread? Why didn’t Hinduism spread?
It is not clear, what time and place is being referred to in this question. When The Buddhist religion began to develop and spread, Hinduism was already a widely practiced religion in Indian subcontinent and in some other surrounding regions. As a matter of fact promotion of Buddhist principles and practices by Gautama Buddha was more like an effort to popularise the a way of life more in line with the principles of Hindu philosophy, by presenting it in form of simple rules of conduct. Thus In India, Buddhism was more of a sect within Hinduism, rather than an religion opposed to Hinduism.
Buddhism also spread to places and people who were not Hindu's because of the efforts of the Buddhist monks following the example of Gautama Buddha to spread the practices prescribed by him for the good of people. This effort was greatly assisted by the support of some powerful people. The most notable among these is the King Ashoka. When Buddhism was thus spreading to people from religions other than Hinduism, the Hindu religion itself did not spread because there were no efforts made to spread it.
As a matter of fact Hinduism is not really a religion. It is a way of life, which also incorporates some religious practices. There is no way available to convert a Non-Hindu to Hinduism. Also there have never been any organised efforts to convert non-Hindus to Hinduism. All the religious movements among Hindu's have been for reforming the religion. Such movement in addition to Buddhist movement includes other movements like Sikkhism, Brahmo Samaj, Arya Samaj, and Radha Soami Panth. Thus though Hindus got divided in many different sects, there was never a significant increase in number of Hindus by conversion from other religion.