Among these ways were: Raids on nominally Roman territories, either to capture land or to settle it; demanding tribute from Rome, a form of bribe not to make trouble; and, finally, sweeping across western Europe and then eastward across north Africa to capture the essential agricultural lands that fed Rome.
In this activity, the Huns were the main instigators (cf. the short but brutal career of Attila), and Rome was regularly distracted by incursions of the Persians and others in the east.
Eventually, Rome simply ran out of resources to deal with all of it, and evaporated into the mists of history.
Anyway, that's Mr. Heather's thesis in a nutshell.