Short answers, pulled from the story.
The central question of the Critique of Pure Reason is: how are synthetic a priori judgments possible? Kant argues that all important metaphysical knowledge consists of synthetic a priori propositions, so if this kind of knowledge is impossible to ground, metaphysics as a discipline is impossible.
In an analytic judgment, the predicate-concept is already contained within the subject-concept, as in "All bodies are extended." In a synthetic judgment, the predicate adds something not already contained in the subject, as in "All bodies are heavy." Kant's innovation was to argue that some synthetic judgments, including pure mathematics, are also known a priori rather than through experience.
Kant wrote that the remembrance of David Hume first interrupted his dogmatic slumber and gave his investigations in speculative philosophy a completely different direction. Hume had argued that the principle of cause and effect cannot be derived from sense experience alone, undermining the rationalist foundations Kant had been raised on. Kant spent at least twelve years working out a response.
Transcendental idealism is Kant's doctrine that all appearances are to be regarded as mere representations and not things in themselves, and that space and time are only sensible forms of our intuition rather than features of objects as they are in themselves. This does not deny the existence of external objects but limits knowledge of them to phenomena as given through sensible intuition.
Kant argues that existence is not a predicate. The word "is" in a declarative sentence merely connects a subject to a predicate without adding a new attribute. Therefore, accumulating predicates in the concept of a most perfect being, including the predicate of existence, can never move that concept from the sphere of inner subjectivity to actual existence. He traces the ontological proof to Anselm of Canterbury, who lived from 1033 to 1109.
Kant compares his critical philosophy to Copernicus's shift from a geocentric to a heliocentric model of the universe. Just as Copernicus allowed that celestial motion is partly due to the motion of the observer, Kant proposes that objects of experience must conform to the necessary conditions of the knowing subject, rather than knowledge conforming to objects. This reversal grounds the possibility of a priori knowledge.