The Roman Empire had been making increasing use of allied Germanic troops for some time. The Romans couldn't stop the massive immigrations, but they could put a vaneer of Romanization on some of the tribes and use them as buffers against other tribes. These Germanics were therefore used to thinking of themselves as belonging to Roman realm in some fashion .... they did take up some measure of Roman law and government.
In fact, most of the Germanic tribes had not come to destroy the Roman way of life, but merely seize the riches of Roman civilization for themselves. We are told by some sources that when some of the Germanic tribes took over, they could be found dressed in togas and living in country villas like the Roman governors before them. Clearly they accepted those aspects of Roman civilization they found comfortable. All they had to do was live in the shadow of the still functioning Roman government in the East, the part of the Roman empire still too rich and defensible for them to carve up completely.
Of course, the Roman influence on these tribes was strongest on the continent and in heavily Romanized provinces. In Britain, the far flung outpost of the empire, most traces of Romanatis soon collapsed under the onslaught of the Anglo-Saxons.