Short answers, pulled from the story.
Nikita Khrushchev delivered the speech on the 18th of November 1956. The event took place before Western ambassadors and envoys from twelve NATO nations and Israel.
The original Russian meaning suggested that the communist system would outlast capitalism over time rather than implying physical destruction. Some modern translators suggest the phrase meant we shall be present at your funeral instead.
Khrushchev stated that he would not bury anyone with a shovel because their own working class would bury them. This referenced Marxist doctrine stating the proletariat is the undertaker of capitalism.
Norris Poulson confronted Khrushchev because many Americans interpreted the quote as a nuclear threat against their nation. He firmly stated that they would not bury each other while defending American values.
Jan Šejna titled his book on communist Cold War strategies after the phrase and Sting included it in his song Russians released in 1985. The phrase also appears in films such as The Courier from 2020 and The Death of Stalin from 2017.