Why did people come to the 13 colonies?

Assuming that you are talking about English citizens during the 1600's and 1700's, the main reason why people left England was for economic reasons. Due to the Price Revolution of the 16th and 17th centuries, many people were forced off of plots of lands because landlords were raising prices because of the massive influx of Spanish gold that was flowing into Europe from Latin America and South America. People turned to the Americas as a chance to regain a way of life that they could no longer have in England. The advent of joint stock companies also allowed many more expeditions to the America's to get underway.

Religion did play a role, but not as much as is commonly thought. The Pilgrims did come for religious reasons, but they were far in the minority. Religious toleration could be found much closer to home in the Netherlands. In fact after the first wave of Separatist Anglicans settled in the America's, religious toleration was actually rare.

Some did come for adventure, but most came for more practical reasons.