It is very fascinating! One way to think about it is to consider something like a foundation of mathematics, a way of formalizing all of (or nearly all of) mathematics with a single overarching theme. So mathematicians study numbers, functions, shapes, and such. What do they all have in common? Is there a way to derive all of them from more basic ideas? At around mid to end of the 19th century people like Cantor, Peano, Dedekind, and many others showed that (virtually) all of mathematics up to that point could be thought of as sets. A number was one kind of set, a triangle another, additions created new sets, as did folding a square, which was also a set. All of it was sets. However, to avoid issues with wanton creation of problematic sets a small list of rules (called axioms) was worked out to prevent paradoxical sets. All sets had to come from one or more rules.
This is normally known as ZFC (after two founders and one of the rules added later). So now they could explore the limits of ZFC, push it as far as it could go, create ever larger and stranger sets (of numbers for example). If you look at the first problem on the Wikipedia page, the Continuum Hypothesis (CH), it says that it is proven to be independent of ZFC. This means that it is possible to create a "universe" of sets that is consistent with ZFC (introduces no contradictions and satisfies every rule) where the CH is false as well as create a universe of sets consistent with ZFC where the CH is true! It took something like 89 years to prove this. This was... incredible. So, if we now ask the question: "Is the CH true?" well we have no answer! We say that the question: "Is the CH true?" is independent of ZFC. We would need to modify ZFC (by adding a new rule for example). We have now found hundreds of problems that cannot be solved in ZFC. Why ZFC, why not other foundations? Well... it works. ZFC is very powerful. There are other foundations but we also know (due to Godel, published in 1931) that they will also have problems that cannot be solved even if we don't yet know what they are. We could add in a rule to ZFC and call the new foundation ZFC+CH, where the CH is true, and this would make the question: "Is the CH true?" have the answer be "Yes" trivially, but we are guaranteed that a new problem will arise that is independent of ZFC+CH (due to Godel)
I hope is an ELI5. A wonderful book on this is called Godel's Proof by Ernest Nagel.