That's a very literal abd misleading reading. The collapse in the West had very specific observable results including:
* A trade crash across the Western empire
* A crash in manufacturing
* Widespread famine
* A retreat from cities and villas
* The withdrawal of the military
* The withdrawal of centralised government influence
* Their replacement by regional warlords
* A huge decrease in civil stability
* A huge increase in petty violence and lawlessness
The millions of people living in the area certainly didn't experience continued expansion and prosperity.
And most writings from Ancient Greece and Rome were lost to the East. Only fragments survived. Byzantium wasn't enough of an outpost to preserve the rest.
What was left in Europe until the Renaissance was a kind of Christianised cargo-cult of the original empire, with most of the knowledge and working features lost. After the Renaissance there was a superficial rediscovery of some of the writing, the architecture, and the history, and it had an influence on modern European culture. But again in a kind of cargo-culty way which eventually turned into something else entirely in every area except politics. (A Roman senator would understand exactly how the US political system works, while being baffled by the technology.)
You can argue that empire building continued elsewhere, but that's a different argument. There were empires in India, China, and the Middle East, but they were distinct and unrelated entities. If anything they've been even more forgotten.