"Woo" is a pejorative label for some types of behavior associated with religion--mysticism, ritual, spiritualism, etc. I was mainly responding to the question in the OP and wasn't trying to explain all aspects of religion and religious behavior. People also use the term "scientism" to describe an exaggerated trust in scientific methodology or just using scientific language inappropriately.
I tend to see religion as grounded in Cartesian dualism--the belief that reality is divided between spiritual and physical realms. Since we directly experience the movement of body parts through pure volition, it is not unreasonable to speculate that external forces are also caused by other "spirits" using the same method to cause things to happen. Animism gives rise to belief in powerful agencies such as demons and gods, which are humanlike beings that are amenable to human influences. They can be cajoled, persuaded, and intimidated. Since they are exotic imaginary beings, people often develop ritual forms of behavior to communicate with them, strike bargains, and offer devotion. Ancient Semites actually formed legal "covenants" with deities. That's how Abraham got involved with Yahweh. He offered Yahweh the loyalty, devotion, and worship that gods seem to desire in exchange for protection, guidance, and good fortune.
Our species is reward/punishment based, that's simply how it has evolved, but why is the woo there? If it is just a social vehicle then lots of people are going about their lives badly misinformed because they certainly think the woo is as real as their fingernails when you and I know it isn't.
Reward and punishment works for all animals with brains, because that is what shapes their model of reality. They learn to avoid punishment and seek reward in all aspects of life. We all have a flawed understanding of reality, but that understanding evolves over time. Sometimes our flawed understanding leads us to do good things for the wrong reasons. As I said earlier, religion plays a variety of roles that can have beneficial effects. The model doesn't have to be perfect in order to have value.
The answer to "Why religion?" must be the human brain. Given two brains, one that knows woo is imaginary and one that knows woo is real, how exactly and physically are those two brains different? They must be quantifiably different to account for the different behaviors. Science is telling us it is in the makeup of the prefontal cortex.
Religious fundamentalism is partly the result of a functional impairment in the prefrontal cortex, new study finds
Religious beliefs can be thought of as socially transmitted mental representations that consist of supernatural events and entities assumed to be real. Religious beliefs differ from empirical beliefs, which are based on how the world appears to be and are updated as new evidence accumulates or when new theories with better predictive power emerge. On the other hand, religious beliefs are not usually updated in response to new evidence or scientific explanations, and are therefore strongly associated with conservatism. They are fixed and rigid, which helps promote predictability and coherence to the rules of society among individuals within the group.
I think that you may have read more into that study than the authors intended. They were talking about a type of brain damage that they believe predisposes people to religious
fundamentalism. It doesn't follow that religion or even religious fundamentalism is generally caused by brain damage. That type of brain damage might also predispose people to other forms of behavior akin to religious fundamentalism, e.g. radical devotion to Donald Trump.
I don't honestly think that the brains of believers and nonbelievers are different. Even atheists can be pigheaded, stubborn, stupid, and just plain wrong sometimes.