The Roosevelt Corollary, as this text is known, was—maybe you guessed it—contained within Teddy "Bear" Roosevelt's Annual Address to Congress in 1905. It's basically tradition to talk about foreign intervention in Latin America in these annual addresses.

Roosevelt invokes the Monroe Doctrine in this speech; however, he also expands on its ideas. He was reacting to then-recent events, when Santo Domingo (present-day Dominican Republic) came to the U.S. for help when it was going to be invaded because of some serious debt. Side note: the U.S. sent in Marines to, uh, help with the situation.

Roosevelt talks a lot about the responsibility of "civilized" nations to be armed and ready to play superhero, with justifications like this:

There are kinds of peace which are highly undesirable, which are in the long run as destructive as any war. Tyrants and oppressors have many times made a wilderness and called it peace… The eternal vigilance which is the price of liberty must be exercised, sometimes to guard against outside foes; although of course far more often to guard against our own selfish or thoughtless shortcomings. (Source)

More specifically, Roosevelt brings up the Monroe Doctrine when he states that "chronic wrongdoing" might require intervention by more "civilized" nations, "and in the Western Hemisphere the adherence of the United States to the Monroe Doctrine may force the United States, however reluctantly, in flagrant cases of such wrongdoing or impotence, to the exercise of an international police power." If only every country would show as much progress as Cuba, Teddy mourns, then the U.S. wouldn't have to get involved.

Roosevelt closes the corollary by explaining that America's adherence to the Monroe Doctrine isn't just about protecting American interests: "we have acted in our own interest as well as in the interest of humanity at large."

Becoming an "international police power" is now more than just preventing colonization (the colonization boom had mostly passed at this point), it's about going in to right wrongs and intervene when atrocity strikes. Although the U.S. has plenty of its own problems, it shouldn't just sit around and let things like the Armenian Genocide just happen.

The Roosevelt Corollary took the ideas of the Monroe Doctrine and adapted them to the 20th century, when the threat of European colonization was pretty much gone, but there were plenty of other things for the U.S. to be concerned with. The corollary expands the scope of the doctrine to not only make the U.S. a protector of the Latin American republics, but a "civilized" policeman for the entire world.