It depends on who you were, but the basic answer is not very much. Some aspects of the Frankish conquest was indeed quite violent, and the Merovingian kings were ruthless enough to make Game of Thrones look tame, but there is a great deal of continuity as well. The Franks maintained the old Roman administration and took on Roman imperial titles. Merovingian charters, like this one here all refer to the king as uir inluster - illustrious man - the 5th century Roman honorific for a senator. I am actually currently doing some work on a late seventh century saint's life, the Passio Praeiecti, which not only contains two men of senatorial rank (Bodo uero et Placidus e sinatoribus uiri), but also refers to the man who martyrs the saint as a 'lictor' (post hec lictor in capud eius gladium uibrauit cerebrumque eiecit). The Franks realized this continuity and stability was to their benefit - you get more taxes than you do spoils - but it was also a fact of numbers. The existing population of Gaul far outnumbered them.
Trade declined as is described in this thread, but it still continued.
Literary culture also went into a decline, and as the old Roman schools closed, education moved into the monastery and the cathedral and became something of an apprenticeship rather than a formalized study. This lead to the rapid "decline" of Latin into what has been called proto-Romance, or a rustica romana lingua, which continued until proto-Romance and Latin were separated from each other during the Carolingian Reforms.
If you have more specific questions, I would be happy to answer them. I would also point you towards this AMA on the Dark Ages.
Sources & Further Reading:
Overview Literature:
Fouracre, Paul, ed. The New Cambridge Medieval History, vol. 1. Cambridge: Cambridge University Press, 2008. A bit dry, but comprehensive, and broken down into sections based on what you would like to read about
McKitterick, Rosamond. The Early Middle Ages: Europe 400-1000. The Short Oxford History of Europe. Oxford: Oxford University Press, 2001. A good introduction
Brown, Peter Robert Lamont. The World of Late Antiquity, AD 150-750. History of European Civilization Library. New York: Harcourt Brace Jovanovich, 1971. If you've never encountered Peter Brown's prose, you're in for a treat; an excellent portrait of the late Antique milieu
Davis, Jennifer R., Michael McCormick, Angeliki E. Laiou, Jan M. Ziolkowski, and Herbert L. Kessler, eds. The Long Morning of Medieval Europe: New Directions in Early Medieval Studies. Aldershot, England ; Burlington, VT: Ashgate Pub. Co, 2008. A compilation of focused studies on the transition to the Middle Ages
Mitchell, Stephen. A History of the Later Roman Empire, AD 284-641: The Transformation of the Ancient World. Blackwell History of the Ancient World. Oxford: Blackwell Pub, 2007. Another decent overview; this one starts a bit earlier
Ward-Perkins, Bryan. The Fall of Rome and the End of Civilization. Oxford: Oxford University Press, 2005. For the argument that the transition from Roman authority wasn't so peaceful
The Late Antique Economy:
McCormick, Michael. Origins of the European Economy: Communications and Commerce A.D. 300-900. Cambridge: Cambridge University Press, 2001. The definitive study on the early Medieval economy
Gelichi, Sauro, Richard Hodges, and Michael McCormick, eds. “Comparing and Connecting: Comacchio and the Early Medieval Trading Towns.” In From One Sea to Another: Trading Places in the European and Mediterranean Early Middle Ages: Proceedings of the International Conference Comacchio, 27th-29th March 2009, 477–502. Seminari Del Centro Interuniversitario per La Storie E Lʹarcheologia Dellʹalto Medioevo 3. Turnhout: Brepols Publishers, 2012. An in-depth study on an early Medieval Italian trading town
Education in Late Antiquity:
Riché, Pierre. Education et culture dans l’Occident barbare, VIe-VIIIe siècle. 4e éd., rev. et corr. Points H195. Paris: Editions du Seuil, 1995. The definitive study on late Antique and early Medieval education; now available in English?
Vessey, Mark. “From Cursus to Ductus: Figures of Writing in Western Late Antiquity (Augustine, Jerome, Cassiodorus, Bede).” In European Literary Careers: The Author from Antiquity to the Renaissance, edited by Patrick Cheney and Frederick A. de Armas, 47–103. Toronto: University of Toronto Press, 2002. A study of the change in literary culture from late Antiquity to the early Middle Ages
Brown, Warren, Marios Costambeys, Matthew Innes, and Adam J. Kosto, eds. Documentary Culture and the Laity in the Early Middle Ages. Cambridge: Cambridge University Press, 2013. A study of early Medieval documentary culture, source bias, and many other things
Other Relevant Literature:
Classen, Peter. Kaiserreskript und Königsurkunde: diplomatische Studien zum Problem der Kontinuität zwischen Altertum und Mittelalter. Vyzantina keimena kai meletai 15. Thessalonikē: Kentron Vyzantinōn Ereunōn, 1977. A classic study on the continuity of law and power between late Antiquity and the Middle Ages; in German
Pirenne, Henri. Mohammed and Charlemagne. New York: Norton, 1939. Outdated, but a classic
Jones, A. H. M. The Decline of the Ancient World. A General History of Europe 1. London ; New York: Longman, 1966. Another classic
Primary Sources (read while waiting for the next GoT episode):
McNamara, Jo Ann, John E Halborg, and E. Gordon Whatley, eds. Sainted Women of the Dark Ages. Durham, N.C: Duke University Press, 1992. A compilation of translated female Merovingian hagiographies
Fouracre, Paul, and Richard A. Gerberding. Late Merovingian France: History and Hagiography, 640-720. Manchester Medieval Sources Series. Manchester: Manchester University Press, 1996. A compilation of translated male Merovingian Hagiographies
Krusch, Bruno, ed. Monumenta Germaniae historica: Scriptores rerum merovingicarum. 7 vols. Hannover: Hahnsche Buchhandlung, 1910. Most if not all Merovingian-related documents extant in c. 1900, in Latin
