Metahphysics as a Science
One of Kant's main goals is to establish philosophy as a true science: something that produces objective and universal knowledge. Some philosophers (Kant groups them under the category of dogmatism) treated philosophy more like religion or art, making abstract claims about invisible essences and using arguments that couldn't be verified in experience—you simply had to believe them. Other philosophers (Kant calls them skeptics, and here Hume is a good example) tried to test philosophy's claims empirically, but then claimed that experience proved that philosophy couldn't make any universal claims about the way things were—we could only have provisional knowledge and guesses, like "the sun will probably rise tomorrow because it seems to rise every morning." For both dogmatism and skepticism, philosophy would never be able to make universal, verifiable claims about what is true.
Kant's goal was to overcome these two, contrasting weaknesses (the unverifiability of dogmatism and the non-universality of skepticism) and to establish philosophy as a science. This is why what Kant calls the "synthetic a priori" is so important for him to establish. If he can show that philosophy produces synthetic a priori knowledge, he would show that it is both genuine, new knowledge about the external world (synthetic) and yet prior to experience (a priori), making it distinct from the natural sciences and a legitimate field in its own right.
The Copernican Turn
One way that Kant described his contribution to philosophy was as the "Copernican turn," and understanding what he means by this can help us why Kant's work was so important and had so much influence on later philosophy. Before Copernicus, astronomers thought that the earth was a stationary point, around which all of the heavenly bodies like the stars, planets, and the sun rotated; the earth was the center of the universe, and our knowledge of the universe was secure because we observed it from a central, unmoving point. Copernicus effected a revolution in astronomy when he suggested that instead of the heavens revolving around a stationary earth, the earth itself was revolving around the sun. This meant that we could explain the movements of the stars by accounting for how we ourselves—on earth—were also moving. Copernicus took the earthly observer's own situation into account in order to provide a better and more objective explanation of the things he observed. Kant makes a similar argument about philosophy: we can get objective and universal knowledge about the external world by taking into account our own position from which we observe it. Like with Copernicus, taking into account the instability or motion of the observer actually allows us to see the object in a more stable, more scientifically-accurate, way.
The consequences of Kant's Copernican turn were huge. Throughout the 19th and 20th century, from Hegel to Nietzsche to Foucault, philosophers and social theorists focused more and more on understanding the subject—the observer—as a way of understanding the world.
