The Renaissance created a number of prominent mathematicians. However, later in the 18th and especially 19th century, Germany and France became the hot centers of mathematical thinking.
This does not apply only to mathematics, but to all ancient culture. It was almost completely forgotten in Europe, and then it was slowly recalled, in the process which is called Renaissance. So, the natural restatement of your question would be: why did the Renaissance happen in Italy first?
It is hard to name one reason. Probably there were several:
First of all, some memories of the ancient culture were certainly preserved in Italy more than in other places in Europe, because Italy was the center of the ancient Roman Empire.
Second, and probably more important, was more intensive contacts with the Eastern Roman Empire, which still existed at the time of the Renaissance, and where some ancient heritage was preserved.
And, finally, Italy in general had more contact with the rest of the world because the international trade was more developed in Italy. This includes Mediterranean countries and the same Eastern Empire. (After all, Marco Polo and Columbus were Italian, and Fibonacci was a merchant.
To a much smaller extent, one can also see a similar process in the Iberian Peninsula, where Europeans had a long contact with Muslims, who also preserved a part of ancient heritage. However, in Spain and Portugal this process was arrested by religious intolerance after the Reconquista.
So the process seems to start on the cultural border (penetration of "new" ideas) but later it develops mainly in the places where intellectual climate is more more appropriate (Germany, England, France).
Analysis got a head start in the work of Italian indivisibilists Cavalieri, Torricelli, degli Angeli, and others. However, the budding analysis was viewed with disfavor by some in the Catholic hierarchy, including many leading Jesuits. This led to a suppression of active math centers in Italy and a graduate decline of the Italian mathematical tradition. The reasons for the clerical opposition to indivisibles are the subject of dispute among scholars. This recent publication by Sherry attributes this to the association of indivisibles with ideas that were seen as contrary to the Catholic interpretation of the eucharist, while this recent publication by Alexander attributes the opposition to an adherence to a Euclidean ideal of rigor.