It's a matter of debate how much knowledge was "lost", or even how to define "the fall of the Roman empire", or even to find a reasonable definition of "knowledge". Nonetheless, some historians have attempted to explore these questions.

First off, in the largest and richest part of the Roman empire, there was no hard-and-fast "fall". The Eastern Mediterranean didn't see a collapse, but instead a gradual transition to the Byzantine era punctuated by various crises. Even many parts of western Europe saw a more gradual transition to a Post-Roman world that began many decades before any distinct "fall" of the Empire, as Germanic elites co-opted existing Roman institutions.

However, in many parts of the western Empire there was a dramatic shift, and in some places it was relatively abrupt. Roman Britain saw perhaps the most abrupt transition to a different, post-Roman era as well as the largest cultural, societal, and economic shifts. Parts of Britain were pretty thoroughly Romanized, with relatively high levels of literacy, technology, and economic sophistication. There is strong evidence a lot of technology, like coins, quality pottery or roof tiles, were available to a large portion of the population and were not reserved for elites. Further, there's evidence that use and production of these technologies ceased abruptly and the standard of living may have plummeted. Large concrete and stone buildings with tile roofs gave way to much smaller thatched-roof wood buildings.

A major cause seemed to be loss of international trade networks caused highly specialized economies dependent on trade with the rest of the Roman world to collapse. Wine, olive oil, and amphora were all integral parts of the economy of Roman Britain but were imported from the rest the Empire, from other specialized region. Localized production of goods simply wasn't able to continue without trade networks and the economy quickly reverted to a lower levels of sophistication. Anyone with specialized knowledge in these fields wouldn't have any reason or way to pass it on to the next generation, so that knowledge died with them.

Culturally and socially, the Anglo-Saxon migrants to Britain were very distinct from the existing Romanized population. Literacy was minimal and they hadn't culturally assimilated to any degree like some other groups on the periphery of the Roman world did. Further, there's evidence that a large part of the existing Romano-British people were conquered, displaced, killed or enslaved by the invaders/migrants to a greater degree than other parts of the Empire. There's debate over exactly how violent this transition is, but all historians agree that there was a major cultural discontinuity. Thus, a lot of cultural knowledge disappeared as literacy plummeted and existing literate people were killed, enslaved, or simply reverted to a much lower standard of living.

Britain is useful to focus on because it was one of the quickest and most abrupt changes in the Roman world. Other parts of the Empire saw much more gradual changes, as Germanic invaders co-opted existing Roman institutions and had often been integrated into the Roman world for decades or centuries beforehand. Nonetheless, many of the causes of Britain's post-Roman decline happened there to some degree as well.

Source: The Fall of Rome And the End of Civilization by Bryan Ward-Perkins