Visigoths in Iberia, great *goth migrations in general.

One series you should seriously consider reading is the trilogy of books about late antiquity by Peter Heather. They are popular enough that getting hard copies of them should be relatively trivial, although I also know ebooks of them are available as well. I recommend Empires and Barbarians: The Fall of Rome and the Birth of Europe as it talks about what you are interested in in particular.

One thing I should note; the Visigoths in Hispania are not strictly speaking the core of the 'Spanish' peoples. The Germans made up a relatively small percentage of all the places in the Roman Empire they carved kingdoms out of, and the bulk of the population were Romanized peoples. Over time, these Germans were absorbed into the larger culture, as happened in Gaul and Italy. Pretty much the only place where Germanic peoples completely pushed out the local cultures was in Britain, and England in specific. Ironically, this is also a good place to consider as a more modern example of how this might have played out, as the Norman kings and Lords of England became 'English' over time, the same as the Franks and Visigoths acclimated to the local culture.

As for a more direct answer to your question, how did that come to pass? Well, the Visigoths in particular were invited by the Romans in the 5th century to settle in Southern Gaul (modern France) like many other German tribes such as the Burgundii and Franks. They were granted nominal tax control over this area in exchange for fighting in the Roman Army. As Roman control broke down further, this went from a very regulated vassal relationship to more of an independent state within the nominal control of the Empire. So they were very close to Hispania. They crossed over there in the wake of another Germanic tribe, the Vandals, rampaging around and basically replaced an absence of Roman authority in the peninsula with their own.

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