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.
