It generally describes diplomacy, saying nice things to others, but actually holding a stick to threaten others. That is to say, you have to pretend to force others to submit.
This sentence comes from US President Theodore Roosevelt. When he said this famous quote, the United States and the world were facing an unprecedented and complex situation.
Expressions with similar meanings to "a gentle word in your mouth, a big stick in your hand":
1. Give a slap in the face and give a candy
2. Use both soft and hard tactics
3. Use both kindness and power
4. Coercion and inducement
Extended information:
Theodore Roosevelt , October 27, 1858 - January 6, 1919), also translated as Theodore Roosevelt, known as Roosevelt, nicknamed Teddy, American military strategist, politician, diplomat, and the 26th President of the United States .
Served as the U.S. Deputy Secretary of the Navy, participated in the Spanish-American War, and made outstanding achievements in the Battle of Santiago, Cuba, and was awarded the title of Hero of San Juan Hill. He was elected vice president in 1900. In 1901, President William McKinley was assassinated by an anarchist. He succeeded him as president of the United States. At the age of 42, he was the youngest sitting president in American history. His unique personality and reformist policies made him one of the greatest presidents in American history.
Reference materials:
U.S. aircraft carrier imitates Roosevelt and "holds a big stick" - People's Daily Online