This is a great question. The main reason why we can look at the Renaissance as a rebirth is because the Renaissance, which took place in Europe starting in the 1400s, was a rediscovery of the classical world. In other words, the Renaissance should be viewed not so much something new, but as a movement that went back to the classical world by bypassing the traditionalism of the Medieval ages.
To underline this point, there was a phrase that was common in the Renaissance, "Ad fontes." In Latin this meant, "back to the sources." In light of this, we can say that there was a impulse in society that wanted to get back to the great accomplishment of the classical world. All throughout the Medieval world there was a fascination with the glories of Greece and Rome. The Renaissance was able to get back and even exceed the accomplishments of the classical world. For this reason, rebirth is an apt way to describe what took place during the Renaissance.