So to my understanding, when the Umayyad conquest of Hispania happened, it was a Visigothic kingdom but then when the Spanish eventually took over, culminating with the Granada war nearly 700 years later they seem to treat it like they were the rightful rulers of Iberia the whole time, and I'm wondering where did the Spanish who "took back" the Iberian peninsula come from and how did they come to believe that the whole thing was rightfully theirs in the first place?
The Spaniards and Portuguese ruling classes were largely descended from those earlier Visigoths and other Frankish nobles. Their armies were made up of professional knights and conscripted peasants brought by the feudal lords as well as the king, as far as I can remember. The moorish side probably did the same in their territories. To what extent christian zeal played a role in their desire to go to war is unclear. However the peninsula was always majority christian since the 3rd or 4th century, even in moorish areas. That's not to say that they didn't prefer their moorish overlords, at least for many dynasties.