You question has books' worth of answers, but I will try to give you an overview of the subject. As a foreword, there is no uniform consensus on this topic and part of my answer will be the most educated guess I can elaborate.

Let's start by saying that the masses of barbarians (taken here as the Greek term for the foreigners) which eventually entered and settled Europe never wanted to destroy Roman territories in any way. Historiography has reconsidered the matter to the point of calling the subsequent polities arisen from the collapse of the Roman system in the West as "post-Roman kingdoms" rather than "Romano-Barbarian kingdoms". This bit is important as it implies an effort of continuation between the newly arrived elites and the institutions and customs which preceded the fateful 476 AD. As a few examples, we know that when Odoacer took Rome and the rest of territories in Italy, he found a functioning circus and hot baths, which continued to operate well into the centuries we call Dark Ages (due to, as you've noticed, a severe lack of informations). When Theodoricus' Ostrogoths settled Italy some decades after, the Roman customs and infrastructures were attempted to be kept running, and partially even in the years after 578 when the Langobards replaced them.

Romanized populations outside the borders were already present well before the years we are discussing. The Franks, the Goths, the Vandals and others had been in contact with Rome for centuries by the years of the Fall of Rome. We know this because several burials which have been found to contain funeral equipment strongly reminiscent of Roman culture. For example, the tomb of Childeric I, Frankish king (d. 481), had a ring with the king drawn wearing the Roman paludamentum and the Latin phrase "Childericus Regis", "King Childeric". In this time, the Franks were still pagan. Similarly, they codified their laws between 507 and 511 in the Lex Salica, written in Latin, similarly to what the Lombards would do in 643 with the Edictum Rothari.

These tribal groups which moved into a more civilized cultural and physical space did undergo a process of change, partially absorbing Roman culture and partially replacing some of its aspects. Religion played a role, as the Franks were the among the first if not the first barbarian population to convert to Christianity, "elevating" them from the rest of the pagans inside the old imperial territories. They brought with them their own customs and laws as we've seen, but never conceived higher culture as none other than Roman, as we know by the documents of judges and notaries other than books of prayer and study which were written only in Latin.
In a couple of centuries, these old barbarians had advanced into a culture very different by those which kept at their same ancestral level well into the Early Middle Ages, as we see with Charlemagne's campaigns against the Saxons throughout the VIII century which were Germanic pagan barbarians and considered as heathen savages by the "civilized" Franks.

To sum it up, the transition between the "barbarian" condition and the Early Medieval populations has no definite explanation. Concurring to it might have been the Christianization process which created an auxiliary net of communications and support alongside what other societal structures there might have been. Surely, the previous Roman cultural layer which was not unknown and rather admired aided in the transition from a tribal society towards a more complex one based on bonds of horizontal and vertical fealty which evolved into the vassallatic system Carolingian Frankia is known for.

Sources:
Provero, L., Vallerani, M., "Storia Medievale", Mondadori Education, Milano, 2016;
Barbero, A., " Barbari. Immigrati, profughi, deportati nell'impero romano", Collana Storia e Società, Roma-Bari, Laterza, 2006;
Gabba, E., Foraboschi, D., Lo Cascio, E., Mantovani, D., Troiani, L., "Introduzione alla storia di Roma", LED Edizioni Universitarie, Milano, 2002.