This is misinformation and a fake quote. What he actually said on that day was:

About the capitalist states, it doesn’t depend on you whether or not we exist. If you don’t like us, don’t accept our invitations, and don’t invite us to come to see you. Whether you like it or not, history is on our side. We will bury you!

“We will bury you,” was a poor translation. What he meant (which is pretty clear from context) was more like, “We will outlast you,” or “We will be there at your funeral.” He never said anything about “We will take down America without firing a shot” or “We will bury America from within” (how do you “bury someone from within,” anyway?).

Khuschev’s strategy towards the West was called, “peaceful coexistence,” the concept being that it wasn’t necessary to take active steps towards overthrowing Western governments, instead seeking to establish an ammenable working relationship, the belief being that the Soviet system would win out in time while the capitalist system would eventually decay and the proletariat of Western nations would eventually radicalize and rise up on their own. You can say that this wasn’t their real belief or whatever, but it was a major contributing factor towards the Sino-Soviet split, and they were willing to stand by that line to the point of alienating other communist countries.

But regardless of whether you agree with that interpretation or not, this quote isn’t just misinterpreting the quote, it’s outright lying about what was said.