It seems like during the 16th Century the Spanish and Portuguese were the only powers colonizing the Americas. I know there were tentative efforts by the French and English during the late 1500s, like the Roanoke Colony. But it seems that once you hit 1600 there's an explosion of French and English colonies in North America and the Caribbean.
My answer is going to focus more on the English side of things. But to be very brief, what happened is that the Spanish wars of the late 16th century, the piracy and privateering that preceded them, and the commerce raiding that happened after the Spanish Armada failed, drove England to seek bases in the New World to attack the Spanish and Portuguese treasure fleets (the flota). What began at least partly as an effort to found naval bases that could watch for the flota in the Florida Straits led to an effort to found more permanent colonies that could take over trade, as the Spanish merchant fleet and its bases in Spain (and sources of seamen) were destroyed by the English over the course of the Spanish wars.
To elaborate on this at a bit more length, the Anglo-Spanish war of 1585-1604 officially started when the English landed troops in the Netherlands to support the States-General during the Dutch Revolt, when the Netherlanders were opposing Habsburg rule. But the decades before had seen an illicit trade between English privateers with Spanish possessions in the West Indies, occasional fights on the high seas between the English and Spanish, and an open battle in 1568 at San Juan de Ulúa. In the 1570s, the English focused their efforts on capturing a Spanish flota (indeed, they kept trying to do the same well into the early 19th century), and Sir Francis Drake famously raided the silver train at Panama, returning with maybe 5 percent of the year's silver, but enough to cause a sensation in Spain and in England. Drake had allied with cimarrones, escaped slaves who had established themselves in the jungle and fought a guerrilla war against the Spanish. Drake's success led the Spanish to fear that with the help of the cimarrones, the cross-isthmus route for silver from South America would be cut off, and indeed there were abortive attempts to find further funding for either an English colony in Panama or a Huguenot colony, but the Spanish subdued the cimarrones in any event by 1579.
In the meantime, though, other French and English raiders followed Drake into the Caribbean and found success in raiding Spanish settlements and capturing merchant ships, though the dream of capturing a flota remained remote. (Spoiler alert, only one was ever captured, in 1628.) There was a plan sponsored by Sir Richard Grenville in 1574-75 which proposed an English colony in the southern part of South America, but the English government refused to grant a charter and the plan was dropped.
In 1577, though, Francis Drake started the most ambitious voyage yet undertaken by an Englishman, taking the galleon Pelican (renamed Golden Hind) into the Pacific with the backing of a joint-stock company. The original plan for his voyage is obscure, but once he passed the Straits of Magellan (only the second European expedition to do so) he sailed into an ocean that the Spanish controlled and where they had never known an enemy. He took booty worth 4,000 ducats from a sleeping Spaniard at Tarapaza, and on the coast of Ecuador captured the Nuestra Señora de la Concepcion laden with 26 tons of silver, among other cargo. He sailed west, crossed the Pacific, picked up a cargo of spices in the Moluccas, and returned to Plymouth in 1580.
Drake's expedition was the second ever to circumnavigate the world; Magellan, who led the first, died along the way. No English ship had sailed south of the Equator until 1555. No English pilot was capable of navigating to the West Indies before 1568; Drake circumnavigated the globe within the next decade. His officially-declared booty was £307,000; the total amount was probably closer to £600,000; his investors made a 4,700 percent return on their £5,000 investment. Queen Elizabeth made £300,000 on her share alone, paying off her entire foreign debt and investing £42,000 in the new Levant Company. It's hard to overstate what a sensation Drake's voyage caused in England, and the possibility of repeating Drake's success in privateering against the Spanish empire and the growing threat that it seemed to pose to England was attractive to Queen Elizabeth.
I'm going to have to stop there for now. What comes next? Stay tuned for the Anglo-Spanish war, the destruction of the Spanish Armada, and the founding of the early English colonies!
I am writing this on my phone so this will be brief. My interest lies in Canadian history so I can't really speak for the British Colonies of the New England states.
It may seem bizarre but fashion and religion were the primary motivators for the colonization of North America in the 1600's.
The early explorations of North America were funded by either royalty or by private enterprise and were sent out to find profits for their backers. They were searching for precious metals and gems like those found in South America and for a Northwest Passage to shorten the sea routes from Europe to the spice trade. Henry Hudson discovered Hudson's Bay and the Hudson River. Jacques Cartier found the St Lawrence River. Neither explorer found the much coveted passage and moved on.
Cartier sailed up the St Lawrence River to the town of Stadacona and kidnapped twoof chief Donacona's sons which he brought back to France. From these two youths they learned of lands and big waters beyond so another expedition was organized and sent back to Stadacona in hopes of finding a route to the Orient. Cartier failed to find this route but he l learned of a land inhabited by Indians who wore European clothes and had many precious metals called Saguenay. He wintered near what is now Quebec city and suffered through freezing weather and starvation. 25 of his men died of scurvy before the Indians taught them the cure of drinking bark and needle teas. When their ships were freed from the river ice Cartier returned to France and brought tales of this new land to France.
War in Europe prevented the French from sending more ships to the St Lawrence for a few years then the king sent off another expedition led by Roberval and Cartier with some 800 men to colonize new France and to find Saguenay. Cartier abandoned the expedition and returned to France with what he believed to be samples of gold but turned out to be fool's gold, iron pyrites. Roberval suffered through a horrible winter and lost many men to starvation and scurvy. This time the Indians were distrustful of the French and their motivations so would not help the settlers. Disappointed the survivors sailed home and France lost interest in North American for a number of years.
They did keep returning to North America but only as far as the Grand Banks because the fishing there was phenomenal. By the start of the 1600's there were over 1000 boats fishing there each year and some of these boats putting into shore for water or timbers would trade trinkets with the Indians in exchange for fur robes.
Back in Europe, felt was used to make hats and the best felt was made from Beaver hides. As the fashion changed hats became larger and more extravagant. The shape of your hat showed your profession and the size and materials making the hat showed your wealth.
This increased demand for felt brought the Beaver to extinction in Europe and revived France's interest in North America. Soon ships that were fishing off the Grand Banks were him home with much more profitable cargoes of fur instead.
The king of France issued charters to companies that wanted to enter the fur trade on the St Lawrence. To protect and enforce his claim to New France each charter had a condition that they bring in a certain number of settlers each year.
There were not a lot of settlers willing to risk a trip to a new world. Ianwas considered a successful trip if only 10% of the passengers and crew died. Then they had to spend a year of hard labour in order to clear an acre of land for cultivation and ran the risk of being killed by warring Iroquois who frequently came into the St Lawrence region to push out the Hurons and gain control of the fur trade for themselves.
But the Jesuits were willing to come over and sacrifice themselves in order to build missions and convert the heathen Indians. They were the ones who came over and established bases in the Indians territories and prepared the way for the later settlements of the French.
Have to go now, but will add more if you're interested.