Great question. I will use the Franks (via Gregory of Tours) to deal with your question as I am most familiar with Frankish material.

Some basic context: in the 5th century, Gaul was Roman. It had Roman towns and Roman citizens (who may or may not have seen themselves as Roman). By the time Sigibert, Guntram, and Charibert were on the Frankish thrones at the end of the 6th century, Gaul was just a part of the massive Frankish kingdom which many claim laid the foundations for feudal Europe.

So, let’s start with a man named Clodio who was a very early King of the Franks (decades before Odoacer had deposed the last western Roman emperor). Gregory tells us that Clodio was ‘a man of high birth and marked ability’, so we may observe that a people whom the Romans called the Franks were already appointing kings before the Roman Empire had completely collapsed. We know very little else about Clodio.

Further down the line (during Clovis’ reign), Frankish kings began to be known as Merovingians. This was based on the idea that the Frankish royal family descended from a mythical sea creature called Merovech (not as strange as it sounds - the modern British monarchy traces their ancestry to the mythical Woden). It was also said that Merovech was a descendant of Clodio, therefore linking this ‘Roman’ Frankish king Clodio to the later Frankish kings. The idea of dynasty was not unfamiliar in these times, there had been great families of emperors and at one point a panegyric even claimed (falsely) that Constantine the Great descended from an earlier emperor of the third century.

So we can see that the Franks appointed a powerful (and proven) noble as their king, but why did they feel the need to appoint themselves a king? The main reason (and what is a major part of the answer to your question) is the collapse of the Western Roman Empire. Gaul had been Roman for 500 years so when everything fell apart, people in Gaul may have felt exposed to exterior threats like that of large tribes sitting on Roman borders. It is therefore no surprise that the Franks chose to go their own way with their own king.

At this point, truly ‘Roman’ Gauls (if they ever actually saw themselves as Roman) may have now felt a danger from the Franks as if they were another barbarian tribe. Gregory tells us that a Roman named Aegidius was appointed commander of the armies in Gaul (possibly by the slowly collapsing Roman government about to be destroyed by Odoacer). Childeric (Clovis’ father) was king of the Franks at this time and was banished by his people, who then chose Aegidius in his place. But Childeric was not one for giving up, and after 8 years in exile returned to be proclaimed king (it seems the Franks did not like Aegidius, the Roman, as their king).

Aegidius died not long after and left a son called Syagrius, who Gregory (oddly) called ‘the King of the Romans’. It seems the Roman army that remained in Gaul had also adopted a dynastic leadership. When Childeric died a few years later, his 15 year old son Clovis ‘replaced him on the throne’ and faced a challenge from Syagrius (like father like son). Clovis was able to defeat Syagrius and become the unchallenged king of the Franks. Clovis was one of the most important Frankish kings, later kings would fail to exercise the same level of control and military might that Clovis was able to express throughout his rapidly expanding kingdom.

Clovis was the king of the Franks, but he was also the product of a Roman province. In what is one his strangest passages, Gregory says that Clovis received a communication from Emperor Anastasius (Eastern Roman Emperor 491-518) which conferred the consulate on to him. Aptly, Clovis ‘stood clad in a purple tunic and the military mantle, and he crowned himself with a diadem’. This is likely an episode that confused Gregory immensely, and one that he probably got a bit wrong. What is true, however, is that Clovis was bestowed an honorific position by the most powerful man in the Eastern Mediterranean. Was Anastasius scared of Clovis, did Anastasius want Clovis’ support in Italy, or was this just a diplomatic courtesy to a powerful European king? It is so difficult to tell, but what we can assume is that life in Francia was influenced by the Romanitas that the area had experienced for half a millennium.

Clovis later converted to Christianity when faced with potential defeat on the battlefield, an episode which is reminiscent of Constantine’s conversion. Gregory describes the newly baptised Clovis as ‘some new Constantine’. Writing many years later, Gregory was still aware of the Roman influence over the Franks, an influence that likely continued until well into the Feudal period. The Church, an institution which was paramount in Francia (and in most later kingdoms), was almost an exact copy of the Roman system that had experienced in Gaul under the Roman Empire. Though the Franks did not have senators and emperors, they were living Roman lives under Roman systems. Courts were a massive part of imperial power since Diocletian’s Tetrarchy (284-305), so it makes complete sense that they thrived under European kingdoms which had less concrete restrictions on noble families.

To answer your question then, Frankish tribes began forming kingdoms as the Roman Empire collapsed around them because practically they needed protection, and in a basic sense there was a massive power vacuum left by the Empire. How they did this is rather unclear, other than that noble families with long-attested power and wealth were able to dominate a new political landscape. This all being true, I hope it is also clear that the Roman influence took a very long time to dissipate from these new European kingdoms, and in some sense never really left in that it made its mark on systems of law/religion/economy/government.

TL,DR: Withdrawal of the Roman Empire made room for others to gain power. Tribes formed together to create kingdoms which flourished under these new circumstances. These kingdoms used Roman-like systems to govern and there was therefore a long transition period from 100% Roman to 100% feudal European kingdom (if that ever got to 100%).

Source: Gregory of Tours, Histories, tr. L. Thorpe (London 1974); II.9, II.11, II.27, II.38.