I understand that for a time the Burgundy family was the royal family of france, but my misunderstanding comes from, if they were conquered by the frankish barbarian forces in the early medieval period, how did they come to power?
I think that the confusion stems from the imbrication of geographic and ethnic terms:
“Burgundians” of the Early Medieval period are a Germanic people that settled in modern Provence & Burgundy in the 5th century. Their ethnonym precisely gave rise to the appearence of the geographic appelation of “Burgundy” as a region within the Frankish kingdom (after they were, as you said, conquered).
Burgundy (or at least a part of it) then became, from the 11th century on, a part of the Capetian sphere of influence. There are two important dynasties of Capetian dukes of Burgundy: the first one created by Henri I, son of Robert the Pious, for his brother Robert, which lasted from c. 1030 to 1361; the second one (sometimes called Valois-Burgundy) created by Jean II of France for his son Philip II the Bold, the most famous, spanned from 1364 to 1482.
So two very influential families “of Burgundy” were indeed taken from the royal family (however, no duke of Burgundy ever became king of France), but these “Burgundians” had absolutely nothing in common with the Germanic tribe of the same name. The only link between the two periods is that the existence of Burgundy as a relatively autonomous region comes from Burgundian settlement. But purely (Germanic) “Burgundian” identity disappeared, as far as we can tell, in the course of two centuries at most.