The proverb "Speak softly and carry a big stick" is often credited to the US foreign policy under Theodore Roosevelt. This phrase refers to a countries military power and how to use it. In this case, "speak softly" refers to the idea that one should not brag about the capabilities of their military. The "big stick" refers to having a strong military.
This phrase fits the idea of imperialism perfectly. Imperialism is the idea of extending a country's power and influence over another country or territory. In this case, the US wants to expand their power by using their military might as a means to do so.
