How does quantum foam refute a flat curvature of spacetime?

Recently I have been reading Brian Greene's book "The Elegant Universe". In the book, he provides the reader with an understanding of both general relativity and quantum mechanics, before ultimately declaring that they cannot work together perfectly. However, the proof used to reveal their incompatibility still has me puzzled.

He states that at a magnitude which is so incredibly small, the curvature of spacetime (which Einstein asserts must be flat) is actually undergoing a rapid frenzy in accordance with the uncertainty principle, and therefor the flatness of spacetime is disproved. However, Greene claims that this frenzy is a result of theorized particles and their antiparticles annihilating to release energy and then subsequently recreating matter over and over again.

First of all, is my understanding of what he is saying correct or am I misinterpreting it? Secondly, why does this disprove general relativity? Einstein states that the curvature of spacetime is flat and only manipulated when in the presence of mass and energy warping it. So shouldn't this microscopic frenzy, which is the result of fluctuating mass and energy, actually work just like relativity describes?

Thank you!