A staggering amount is an astonishing, astounding, stupefying amount. Anything staggering blows your mind.
If you know that stagger means to stumble around uncertainly, then you're close to the meaning of staggering: this is a word for things that are so hard to believe you might fall down when you hear them. If the President gets shot, that's staggering news. If there's a major earthquake, that's staggering. Alien life contacting the Earth would be extremely staggering. Anything that knocks your socks off or makes your mind reel is staggering.