The previous post was extremely thorough. I would also add that the need to solidify the economic interests of the United States abroad helped to further Roosevelt's claim of imperialism. As European nations began to further their hold on other nations as colonies, American interests also began to seek diversification. In the increase of American economic holds on nations through loans, assembling spheres of influence which would protect economic interests, and allowing business aims to drive governmental policy, Roosevelt's advocacy of increased imperialism can be seen as an attempt to further business growth within and outside of America.
Excellent answer above. I have a few additional points to consider where Roosevelt was concerned:
1) There were many other empires at the time competing for the world stage, including Germany, Japan, Britain and France, so there was a race to colonize Asia starting right about the turn of the century. Teddy Roosevelt wanted to stake out a position there for the United States.
2) As the world land grab was taking place, Roosevelt wanted to assert our authority over North and South America, in part to increase our wealth and power, and in part to prevent European empires from doing so in the Western Hemisphere. To back this up, he dusted off the 1823 Monroe Doctrine and added the Roosevelt Corollary, which argued that the Caribbean was an "American lake".
3) America in TR's time was the land of segregation and anti-immigrant movements and laws. For him to believe in white supremacy overseas was not a surprise, and made him more popular at the polls.
There were a number of reasons for this. They boil down to the idea that Roosevelt wanted the US to be more powerful and thought that empire was a good way to accomplish this.
There were a few different kinds of power that he had in mind. First, there was military power. He wanted the US to have bases from which it could project military power around the globe. Second, he wanted economic power -- he wanted to be able to trade with other countries.
Finally, Roosevelt was interested in the US maintaining its power for racial reasons. He was concerned that the white race should remain dominant over other races and thought imperialism could help with this.