For example, we define 'what a set is' or 'what natural numbers are' using a bunch of assumptions about how they should behave. Those assumptions are all there is to the 'world or sets' or 'world of natural numbers'. So we cannot assert that a certain behavior of 'sets' or 'natural numbers' is true if it does not follow from our assumptions. The Incompleteness theorem seems really unremarkable in this sense. Mathematical structures are thought up by humans. Suppose we think of a universe where movement is not possible and we set up the rules for the behavior of the member particles of the universe. Then Incompleteness theorem is like saying "the thorey does not describe movement behavior of particles", even where there is no such thing as a 'movement behavior' in our thought-up world.
What am I missing something? Why is this theorem remarkable?
Lots of questions very similar to this one have been asked on this site, but at the moment I can't find an exact duplicate. So here goes:
I think the key observation (I'll make another one below) is the following:
We can dodge the philosophical issues here entirely.
One can dodge the entire philosophical discussion by noting that there is a "purely formalist" version of Godel's incompleteness theorem, which makes no reference to truth at all. Namely:
More snappily, $Q$ is essentially incomplete. (This can actually be improved substantially - $Q$ can be replaced by even weaker systems, and "containing" can be replaced by "interpreting" - but let's ignore this for now.)
This is still fairly unsurprising to a modern audience, who are familiar with computability theory and recognize that this is more-or-less the incomputability of the halting problem in disguise, but in $1931$ that notion wasn't yet well-developed. Without it, essential incompleteness is in my opinion quite counterintuitive - why shouldn't there be a "simplified" arithmetic system (regardless of whether it matches our intuitions about the natural numbers, or - adopting a Platonist standpoint - whether it is true of $\mathbb{N}$)?
OK, now let me push back against the philosophical assumptions in the question itself. As noted above, this isn't necessary to defend the meaningfulness and interestingness of incompleteness, but it's still worth doing. The point is:
Not everyone agrees with you here.
This question adopts a formalist stance ("Those assumptions are all there is to the 'world or sets' or 'world of natural numbers'"). This is not universal: there are for example Platonist, or "Platonist-for-$\mathbb{N}$," mathematicians out there, regardless of how strange that may strike you. To these mathematicians your dismissal of incompleteness is largely beside the point.
Now you might respond that you don't care since Platonism is "obviously silly" - there are also plenty of mathematicians who take this stance - but I don't think that gets you out of this entirely. Even from the standpoint of formalism we can still argue that one crucial feature of mathematics is its universality despite philosophical differences (a Platonist and formalist can work together to prove the same theorems). From this perspective the fact that the incompleteness theorem is compelling from at least one philosophical stance (Platonism) means that it can't be rejected as uninteresting even from a radically different stance (formalism). Again, this may seem weird but there are serious mathematicians who hold it (at least one, to be precise :P).
But, as noted above, we don't even need this.