Speaking specifically for American colonies but also of colonialism in general, they came for profit from resources, at first being gold and silver but later being trade goods such as fish, fur, and wood, which later evolved to more profitable crops like tobacco, indigo, and rice but not until after the formation of the colonies.

More can always be added, of course, but I wrote about some profit motivations for the joint stock venture companies that funded colonizaton [here] (https://www.reddit.com/r/AskHistorians/comments/gjj4kk/in_the_early_days_of_colonialism_in_north_america/) (In the early days of colonialism in North America by Europeans, the driving factor for founding colonies was profit. As an investor of one these expeditions, where would I expect my profit to come from? How was money made by founding a colony in North America?) and why they used these companies [here] (https://www.reddit.com/r/AskHistorians/comments/il8cmd/why_were_the_early_us_colonists_apart_of/) (Why were the early US colonists a part of companies? What was the reason for calling these groups companies?), to which I'll briefly add that the Spanish essentially did the same - conquistadors were not employees but were chartered individual "armies" able to keep a portion of their own profits as payment, hence their gold fever.

I answered questions about who started that trend for England and how [here] (https://www.reddit.com/r/AskHistorians/comments/hjvktv/what_led_humphrey_gilbert_to_colonize/) (What led Humphrey Gilbert to colonize Newfoundland? And what were the results of said colonization in terms of subsequent explorations?). One interesting tie in I'll edit in - those profit seeking folks in 1607 that formed the Virginia Company were split in half. The second in command for the northern colony (from the Virginia Company of Plymouth), Popham Colony, was Raleigh Gilbert, who was invested in the endeavour. He was the son of Humphrey. Popham died and Gilbert inherited a large estate back home, and those are the two biggest reasons the colony was abandoned.

Happy to clarify or expand on anything that isn't answered in those, what the specific contracts with the joint venture companies looked like, or to answer any followups that those posts inspire.

A great source for Atlantic colonization is American Colonies by Alan Taylor, which covers post Columbus to the 1800s very well and touches on the motivations for multiple "companies" within multiple nations all vying for land and subsequent wealth in the New World.