First, I think its important to be pedantic and out that only the Western Roman empire "fell" in the time frame you are likely referencing and that to some degree points towards the likely answer: The roman empire was eventually split into two parts with one emperor ruling the west and one ruling the east. A major problem for the west was that they had a disproportionate share of the expenses in having a military presence in the far flung provinces yet most of the wealth was in the east.

This economic pressure on the west was untenable and as the centralized government weakened there was a clear shift to local leaders. It important to note though that from the participants at the time the roman world just didn't fall/end. From their perspective the "roman world" mean the "civilized world" and even after local military leaders became kings (such as the franks and burgundians) they still saw themselves as part of the roman system and formally pledged loyalty to the eastern roman empire. Furthermore, the long standing senatorial families who were a large transmitter of roman culture survived the western "fall" to a great degree and much of the roman social system was transmitted to them as the local heads of the catholic church (bishops). Its important to remember the church was a roman institution and had become a central component of roman society and culture.

Its important to keep in mind the context that while there was all this continuity there was also a lot of change in the west. More than anything long distance trade was dramatically reduced so material culture took a step back in many ways and also outside groups found it easier to raid different regions.

A caveat to all this: I am most familiar with the Gaul province and my answer is from that perspective. If someone was say a northern African specialist they may have a bleaker view of the matter.