There are three main reasons why the United States engaged in colonization in the late 1800s and early 1900s. One reason was economic while the other two had to do with power in the international arena.
One reason why Americans wanted to take colonies was to make the country richer. Americans believed that taking colonies would give them access to more resources. For example, by taking Hawaii, the US got access to the sugar that the islands produced. In addition, colonization would give them more markets in which to sell. Countries that the colonized would be more likely to buy manufactured goods and other products from the US. In these ways, colonization would boost the US economy.
In the international arena, colonization was seen as a way to increase American military power. The American naval thinker Alfred Thayer Mahan said that countries could only be strong if they had naval power. By taking colonies, the US would get itself more places to put naval bases like the one at Pearl Harbor. This would enhance its ability to project military power around the world.
Also in the international arena, having more colonies would make the United States more of a world power. All of the major European countries had colonies in various places around the world. If the US wanted to be seen as a world power on a par with those countries, it had to have colonies of its own.
Thus, there were at least three main reasons why the US engaged in colonization in the late 1800s and early 1900s.