During the Renaissance, European banks and businesses stopped using Roman numerals in their record-keeping and switched to Arabic numbers. What were the reasons for this change? How did it affect the way Europeans did business?

First of all, they are properly called Hindu-Arabic or Indo-Arabic numerals, because they originated in the Indian subcontinent, were adopted by the Arabs and then reached Europe.

And it was actually quite a struggle to get Europeans to favour them over the Roman numerals they had been using for so long, eventhough the advantages in ease of arithmetic, etc, seem so obvious to us now. Witness this woodcut from 1503 (Gregor Reisch, Margarita Philosophica) in which Arithmetica is shown instructing an algorist (left) and an abacist (right).