This question is actually the reverse of what should be asked about this Theorem: why is it so famous within mathematics? Answers by mathematicians to this question assume their conceptual math problems are at the heart of this Theorem and thus ridicule its application to conceptual problems of formal logical explanations for anything else. This question as most answers to your question by mathematicians miss the point of both Godel’s Theorems specifically and of logic generally. Godel was a logician —- a classical logician —- not a mathematician. For Godel and for most logicians, logic is a rigorous formal language that avoids the vagueness of natural language. For Godel, logic was supposed to provide a formal language by which one can talk about reality directly and clearly by translating natural language sentences into formal language sentences or, for science at least, by eliminating the need for natural language sentences. For Godel, even mathematics could be translated into a formal language of logic. As a classical logician, Godel was trying to unite logic and mathematics; as for classical logic, the three classical axioms of logic consisting of identity, non-contradiction, and excluded middle are implicitly assumed so they would have to be assumed in any unity of math and logic.
For logic to be used as such a universal formal language, it runs into the same self-reference problem as do natural languages. The simplest example is usually the sentence: “this sentence is false”. If false, it is true; if true it is false. However a better example is: “the lowest positive integer not definable in under sixty letters”. Given there are a limited number of characters in any language alphabet formal or otherwise, there must exist such an integer; however, once you find it, it is defined by the letters in this sentence which total <60. So, even this sentence about numbers is always either incomplete or inconsistent.
What Godel did was express in mathematical logic this problem of self-reference in which self-referencing sentences always come out either incomplete or contradictory. He used mathematical logic because it is rigorously defined and lacks most of the vagueness and indeterminacy of natural language.
In logic and in mathematics, this problem of self-reference can be avoided by creating meta-languages that talk about object languages instead of themselves. Logicians now know that if they assume the right axioms or assumptions, they can pretty much create a formal logic that says anything they want it to say. So, for example, in para-consistent logics, Godel’s Theorem is not even really a theorem because contradictions do not mean anything beyond a specific sentence at issue. Mathematicians can avoid self-reference by the same means; i.e., creating a “True Arithmetic” that nominally has an infinite chain of axioms or meta-languages.
However, regardless of how meaningful the concept of “infinity” is to the mind, machines and computers and anything in the real world dependent on a formal logic cannot have an an actual infinity of anything in their logic. Machine/computer logic cannot reason with an actual abstractly imagined “infinity” of axioms, things, relations, or whatever as your mind does and is doing when it creates any concept of self-reference requiring the classical axioms —- or not requiring them for that matter. A formal logic cannot create another formal logic without knowing or having truths outside its formal logic it can use to create another formal logic unless it is contradictory in which case it can create an infinity of other formal logics: it is either incomplete or inconsistent. Thus, colloquial summaries of Godel’s Theorems are accurate regardless of how mathematicians complain about their vagueness as are all natural language statements. The fact that the mind not only can understand and operate but prosper with both incomplete and contradictory logics so as to create conceptual realities it cannot talk about categorically differentiates it from the body, matter, or whatever words you use to differentiate active reality from passive reality. “It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, we had everything before us, we had nothing before us, we were all going direct to Heaven, we were all going direct the other way …”, so on and so forth for the mind and its creativity.
