First of all, it is not really accurate to say that religion began in the Neolithic. It is probably true that organized religion began at that time, but we cannot say that religion in general had its origins then. People were certainly religious in some ways long before the Neolithic.
Why, then, did organized religion begin in the Neolithic? The major reason for this is agriculture. Agriculture changed people’s ways of life in very important ways. The most important way in which people’s lives changed, for the purposes of this question, is that they came to live in large settlements.
Before agriculture, almost all peoples of the world were nomadic. They were hunter-gatherers who typically had to move around to follow their food supplies. Hunter-gatherer societies cannot be very big. They consist of small groups, all of whom are very familiar with one another and many of whom are related by blood. Such societies are typically very egalitarian with no elaborate government or social structure.
With agriculture, this changes. People start to live in one place permanently. Societies become much larger and people no longer know everyone in their group well. A hierarchical government arises. In such a society, religion is needed for two reasons. First, religion provides a way to keep society peaceful. Religion helps to keep people in line by giving them more of an incentive to behave in the “right” ways. Second, religion is needed to help prop up the government. Governments are claiming the right to rule people and to take some of their possessions through taxes. Religion helps allow this to happen because religion can be used to explain why the rulers have the right to rule.
Thus, organized religion began during this time because farming led to large-scale sedentary societies.