Not only in the Iberian peninsula, but also in general, the Empire fell little by little. That is why you cannot give a special date to the fall of the Roman Empire. You can rather talk about a successive de-Romanization of the Western Roman Empire. Due to the fact that the central authority was weakened in Hispania during the 3th and 5th century, this process could take place. However, there was at the same time a process of Romanization of the Germanic and Hunnic tribes that settled down on both sides of the limes (which used to be the frontier of the Empire along the Rhine and Danube rivers). The Visigoths, for instance, had to convert to Arian Christianity around 360. That was even before this tribe was pushed into imperial territory due the expansion of the Huns. It was in the winter of 406, when the Germanic Vandals and Sueves, as well as the Sarmatinan Alans took advantage of the fact that the Rhine was frozen. Thus they could invade the empire with force. In 409, three years later, these Germanic tribes reached the Iberian Peninsula and divided the Western parts. These parts correspond more or less to today's geography of modern Portugal and western Spain. However, the Visigoths, arrived in 412 and founded Toulouse, the ancient Visgothic kingdom that is today located in Southwest France. Little by little, this tribe could expand its influence on the Iberian Peninsula. That is why the Alans, as well as the Vandals, left the territory and moved to North Africa. Overall, these two tribes did not really have an influence on the Hispanic culture. The Visigoths changed their capital from Toulouse to Toledo and the tribe hit its peak during the reign of Leovigild.
Fortunately, all those different people living in the Iberian Peninsula, in contrast to many other regions such as Britain, Lombardy or Germany, were all interested in classical culture. That is why Spain's cultural offer is unique. The period of Visigothic rule saw the spread of Arianism briefly in Spain. In 587, Reccared, the Visigothic king of Toledo, had converted to Catholicism and eventually put an end to dissension on the question of Arianism and launched a movement in Spain to unify the various religious doctrines that existed in the land. In the yaer of 546, the Council of Lerida constrained the clergy and extended the power of law over them under the blessings of Rome.
The Visigothic did not really merge with the Spanish population, as they preffered to be seperate. That is why today, neither the Spanish nor the Portuguese language have many words with an origin in the Visigothic language. The most visible effect of the visigothic period was the deurbaisation as a lot of people moved to the countryside. Even when France and Germany had to suffer from famine, the Iberian Peninsula enjoyed more or less prosperity. However, the Visigoths were not interested in contributing to the welfare of their people, or a better infrastructure of the state. This was one of the main reasons of the downfall of the Visigothic Empire, as they could not count on the loyalty of their subjects when the Moors arrived in the 8th century.