Some good reading on this subject is Thomas Charles-Edwards' After Rome and Robert Bartlett's The Making of Europe.

The essential answer is that, really, these types of systems existed prior to the Roman Empire, as well as contemporaneously with it, both externally in "barbarian" kingdoms and internally in areas settled by foederati in the later Imperial period. When the centralising bureaucracy of the Empire began to fail and eventually collapse entirely, all that happened was that populations began to fall back to the old models which had maintained their society previously, or re-establish them within the bounds of new polities. Most often, this meant deferring to a tribal heirarchy, such as the Anglo-Saxons in England or the Franks in Francia. The payment of taxes to a centralised authority moved to the rendering of goods or service to a chieftain, warlord or local king.

The key difference is the legacy of the Empire which remained in infrastructure, language, literacy and crucially the Church, which allowed far more substantial and bureaucratic heirarchies to form than those which had existed prior to Rome. Compare, for a good example of how important these legacies were, the pre-conversion Anglo-Saxon kingdom of Kent, to the eventual post-conversion kingdom of Wessex.

The 'Feudal system' as we recognise it, therefore, is largely the extension and reproduction of earlier pre-Roman societal models, but hybridised with the organisational legacy of Rome, allowing the system to be replicated to a much more sophisticated extent over a far wider area.