How did the Hindu religion emerge from changes in culture or society?
I would suggest that one of the most significant elements regarding Hinduism's emergence came from its integration with Western values. Hinduism was indigenous to the Indian subcontinent. It remained fundamentally distinct from the "Big Three" of Western Religions. While Western religious experiences like Judaism, Christianity, and Islam emerged, Hinduism remained in the Indian subcontinent. As Western modes of reality entered into the Indian subcontinent, Hinduism did not repel Western religious expressions. One of the most distinctive aspects within Hinduism's emergence was that it did not repel such change, but rather sought to integrate these changes within its expression.
Upon the West's interaction with Hinduism, significant change in culture and society was present. Western appropriation of consciousness was fundamentally different than what Hinduism offered. Western notions of personalized expression in the form of affirming individualism, binary modes of understanding, as well as distinct judgments related to salvation and condemnation became part of the Indian subcontinent with the expansion of European colonization.
These realities posed significant changes to the culture and society within which Hinduism existed. Yet, Hinduism was seen as being able to integrate these changes in providing a view of the world. Thinkers like Swami Vivekananda were essential in suggesting that Hinduism embraced spiritual tenets that underscored all religious experiences:
No religion on earth preaches the dignity of humanity in such a lofty strain as Hinduism, and no religion on earth treads upon the necks of the poor and the low in such a fashion as Hinduism. The Lord has shown me that religion is not in fault, but it is the Pharisees and Sadducees in Hinduism, hypocrites, who invent all sorts of engines of tyranny in the shape of doctrines of Pâramârthika and Vyâvahârika.
Thinkers like Swami Vivekananda argued that Hinduism was a religion that merged opposites together. It provided a universal oneness while enabling individuals the ability to choose, thereby appropriating aspects of both Western and Eastern experiences:
The fault with all religions like Christianity is that they have one set of rules for all. But Hindu religion is suited to all grades of religious aspiration and progress. It contains all the ideals in their perfect form. For example, the ideal of Shanta or blessedness is to be found in Vasishtha; that of love in Krishna; that of duty in Rama and Sita; and that of intellect in Shukadeva. Study the characters of these and of other ideal men. Adopt one which suits you best.
The changes in Indian society in which Western notions of identity were becoming rapidly accepted were realities that Hinduism merely appropriated into its own vision. As a result, Hinduism was "bringing to the West teachings which have become an important cultural force in western societies, and which in turn have become an important cultural force in India, their place of origin." This helps to explain how Hinduism emerged from changes in culture and society. Its tolerance and willingness to assert universality in the face of contingency helped Hinduism to emerge from changes in culture and society.
All religions, including Hinduism, are primarily means of shaping the behavior of individuals of society. On one hand, religions shape the nature of society and culture. On the other hand, new religions emerge and old ones change in response to the changing nature and needs of society and culture.
There is no short answer to, "How did the Hindu religion emerge from changes in culture or society?"
Hindu religion is thousands of years old, and has been evolving all alone. Some of the oldest texts of Hinduism - including the "Mahabharata" and the "Ramayana" - contain many stories suggesting changes in the social practices of that time.
Gautam Buddha who lived about 2500 years back was also a reformer of Hindu religions. Outside India, the sect of Hinduism following teachings of Buddha, ultimately got merged with Hinduism. Although, outside India, where Hindu religion did not exist, it survived as a separate religion.
About 500 years back saints like Kabir sought to reform practices of Hinduism. The efforts of one such reformist - Guru Nanak - resulted in formation of one new major religion - the Skhism. Anothe major reform in Hinduism started by Swami Dayanand about a century back resulted in formation of a new sect called "Arya Samaj" For all practical purposes this sect has been fully integrated in mainstream Hinduism.
Thus Hinduism has been changing and being incorporated in its social and cultural practices with regularity.