Roosevelt used the "big stick" because he believed he had it to use. The US had proved a few years before that its navy could stand up to Spain and that the army and marines could hold its own its own in small-scale land wars. He believed that the US should continues its policy established by the Monroe Doctrine decades before to protect American countries from foreign intervention or take-over. The problem in his day was more from economic threats from foreign countries trying to collect debts than from dreams of colonization, but Roosevelt saw a parallel.
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Theodore Rosevelt's Big Stick Diplomacy was to increase US intvestments in businesses. And to tell other countries that we have a strong military "big stick" so we can speak softly which means we don't need to effort much then we can get what the US wants.
Roosevelt believed the united states had a responsibility to "civilize" other nations