Why did Theodore Roosevelt, Elihu Root and Henry Cabot lodge advocate American imperialism?
"....if we are to hold our own in the struggle for naval and commercial supremacy, we must build up our Dower without our own borders"--Teddy Roosevelt
There were a number of motives for the interest in American imperialism in the beginning of the Twentieth Century. Of great import was the interest to strengthen the military position of the United States. Naturally, Latin America sits between the Atlantic and Pacific Oceans and acts as a sort of geographical bridge. The United States was very interested in expanding its influence in the Pacific. In order to achieve this, it had to have naval bases throughout both coasts. Having allies in Latin America could assist in developing a naval power for both oceans.
"I would rather see the United States respected than loved by other nations." Henry Cabot Lodge
Another motivation for American imperialism was good 'ole fashion nationalism. Many politicians, including the three in your question, felt that to be recognized by Europe as a world power, your country needed to acquire an empire. This was the way of the world between about 1850 until the end of World War II. There was obviously a component of racism that went into this methodology, as white Europeans did not feel that nonwhites could adequately govern themselves. Many Americans used this argument to suggest that European powers were actually providing a service to their subjugated peoples.