Newton's contribution was in explaining how the planets move around the Sun, not that they did. He did this through his law of universal gravitation, which states that every particle in the Universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This law explained the elliptical orbits of the planets and the way they interacted with each other and the Sun.
So, while Newton didn't discover that the planets go around the Sun, his work revolutionized our understanding of how they do it.