First of all there was the question of comets. Comets appeared, and seemed to follow paths like the ones that planets did. Why?

Second there were moon systems. Starting with Galileo, people knew that Jupiter had moons. They were able to see that Jupiter's moons seemed to follow a Keplerian rule of their own around Jupiter with different constants. Coincidence? It was natural to try to generalize. Particularly since we had our own Moon that didn't fit the system.

Third, Kepler's laws were not perfect. While they are fairly accurate, they don't take into account perturbations in the orbits of the planets from other planets.

Fourth, gravity is all around us. It was one thing to say that Venus moved around The Sun. It is quite another to explain why Venus doesn't fall on our heads instead.

And the last reason was the hope that a better theory would explain more phenomena. And it did. Newton fit all of the above together while explaining tides, the precession of the Earth, and various other perturbations.

An interesting side-note. It turns out that if we had a binary star but were far enough away, that The Earth could readily have survived as long as it has. However figuring out orbital mechanics would have been insanely hard. The simple perioodicities which lay behind theories from the Greeks through Kepler only happen when a single gravity source dominates. Arriving at the same conclusion would be far harder with 2 interacting gravity sources acting on everything else.

Cheers,
Ben