Newton formally linked the ideas of mass, acceleration, and weight, as well as the idea that weight and force are the same thing, and also linked force to the idea of changing momentum. He did all of this within a unifying theory of mathematics that relied on his fancy new tool that we would call calculus.

Now, there were other scholars in his day that had achieved bits and pieces of this already, but he brought it all together under a single unifying scheme and added significant portions of mathematical analysis to it (that allowed us to make novel predictions about the world and other worlds), so he's the one we remember. (It also helps his fame in the Western world that he wrote it in Latin, which was the scholarly gold standard of the day)

But the truth is we more or less understood how gravity works (mechanically, that is) since roman times, and newton didn't answer the "why" for us really anyway. What Newton really did was give us a formal mathematic definition of gravity that could be applied robustly to most physical situations humans can observe under one very large tent of mathematics. He allowed us to agnostically apply one set of laws to a whole host of situations that were not obviously connected. This, more than anything else, was his contribution to science.