The name "big stick" diplomacy stems from an African proverb TR admired: "Speak softly and carry a big stick." It basically means to not be aggressive at first but to threaten to use violence if necessary. Here's one example:
When TR was President, Panama was part of Colombia. TR wanted to build a canal through present-day Panama, but when he asked Colombia for permission to do so, he was denied. At the time, Panamanians wanted to break free from Colombia, so TR helped them to do just that. Once they had successfully broken off, Panama let him buy a strip of land to build his canal for 10 million dollars. He continued on to build and eventually finish the canal (the Panama Canal).
TR's drastic actions of aggression towards Colombia for not getting his way are a prime example of his "big stick" diplomacy.