Why is neil Armstrong's quote important?

Well, it just about summed up a generation worth of triumph,tragedy, successes and failures.

"That's one small step for man, one giant leap for mankind"

It also compares the literal to the figurative. Literally, Armstrong made a small step off the LEM, figuratively that one small step was as important as inventing the wheel or discovering fire, because it was SUPPOSED to "leap mankind" into the next generation of humanity, that involved space exploration, regular visits to the Moon, Mars and beyond, new science and engineering, among many other things.

The real question is "why didn't that happen?"

I hope I am not answering a homework assignment for someone here, Neil Armstrong, NASA, the Apollo Project are all topics of amazing wonderment, I suggest you google around or visit NASA.gov for more of the story of the greatest event in humankind, ever.