The main factor that led the European powers to spread their influence across the rest of the world was, put simply,
that they could.
The European powers-France, England, Spain, Portugal, etc.-had resources and technology that the rest of the world lacked.
The tribal peoples of Africa and the Americas seemed to devote most of their time and resources to competing with one another.
Unlike Europe, they still lived in small clans and tribes, much like city-states that were completely autonomous. This system
doesn't lend itself well to technological advancement, as much of their time and resources necessarily were devoted to meeting
basic needs such as food and shelter, and competition among tribes was therefore fierce.
In Europe, by contrast, the medieval serfdoms had long since evolved into monarchies made up of many cities and provinces,
with political power spread throughout complex hierarchies. With many of their basic needs met, the aristocrats of Europe
could therefore devote much more time and attention to the world beyond their borders.
To what end, and for what purpose? There was naturally competition among the nations themselves, but they had long since
reached an equilibrium that prevented any one nation from presenting much of a threat to any other. That left the "undeveloped"
continents of Africa and the Americas and even, to a degree, Asia.
Resistance to European encroachment onto their lands was, mostly, futile, and the continent of North America was especially
tempting to the colonial powers. It was rich in undeveloped resources and, unlike other continents, had a favorable climate
to the Europeans.
So, why spread their influence to these places? Because they could. The English could, and they knew it. The French could,
and they knew it. What's more, they knew that the English could, and vice-versa. The temptation to exploit these areas and
spread their influence, to gain more wealth and power, was obviously great; it was tempered, though, by the fear that someone
else would get there first.
Another factor may be more basic, and back in those days a lot of people liked to believe that they were better than
everyone else and superior to those around them. This came into play a lot with religious groups, of course, but there was
a cultural component to this as well, that many Europeans believed what they were doing was civilizing the savages, converting
the heathen and, basically, saving them.
It may work the other way around. This idea of spreading their civilized ways, their superior technology and culture
and art and music and, yes, religion, may have been justifications that they used after the fact. It may be simple arrogance
to say that because they were more powerful, they were necessarily superior, and these arguments may have been used after
the fact to justify what they did; after all, more often than not, they didn't try to civilize anyone, they just started killing
them off.