The most oft-quoted of Gödel's philosophical works might be Is Mathematics a Syntax of Language? It is not available online, unfortunately, but excerpts are read in this YouTube video. Tait wrote a survey Gödel's Unpublished Papers on Foundations of Mathematics, which is also freely available. For Gödel's metaphysics and epistemology see Ternullo's Gödel's Cantorianism and Solomon's On Kurt Gödel's Philosophy of Mathematics gives another overview.
Young Gödel was initially associated with the positivist Vienna Circle and was courted by Carnap himself. On this period, which led to the incomleteness theorems see Goldfarb's On Gödel's Way In: The Influence of Rudolf Carnap. But he already had misgivings, and his growing fascination with Leibniz shows strong Platonist leanings, see Why did Gödel believe that there was a conspiracy to suppress Leibniz's works? Gödel's Is Mathematics a Syntax of Language? written in 1950-s is a repudiation of Carnap's positivist philosophy of mathematics (nutshelled by the title), and the philosophical alternative that attracted Gödel in the later years was Husserl's phenomenology:
"Now in fact, there exists today the beginning of a science which claims to possess a systematic method for such a clarification of meaning, and that is the phenomenology founded by Husserl. [...] But one must keep clearly in mind that this phenomenology is not a science in the same sense as the other sciences. Rather it is [or in any case should be] a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us."
The idea of fallible extra-sensory "perception" of mathematical objects, which is how Gödel interpreted Husserl, was later developed by some analytic philosophers, like Maddy:
"It should be noted that mathematical intuition need not be conceived as a faculty giving an immediate knowledge of the objects concerned. Rather it seems that, as in the case of physical experience, we form our ideas also of t’ose objects on the basis of something else which is immediately given. [...]
Evidently the "given" underlying mathematics is closely related to the abstract elements contained in our empirical ideas. It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality."
"Purely subjective" is not how Kant would put it, but for Gödel anything coming from the subject is inferior, and this covers not only Kant but Carnap and Wittgenstein too. Late Gödel does not hide his philosophical sympathies on the side of Plato, Leibniz and Husserl:
"I believe that the most fruitful principle for gaining an overall view of the possible world-views will be to divide them up according to the degree and manner of their affinity to or, respectively, turning away from metaphysics (or religion). In this way we immediately obtain a division into two groups: skepticism, materialism and positivism stand on one side, spiritualism, idealism and theology on the other."
