Superb question! Interestingly, my math professor asked the exact same question after teaching Galois theory and stated that he does not have a good answer himself. Let me try to give sort of an answer. :)

We have fingers. These we can count. This is why we are interested in counting. This is gives us the natural numbers and why we are interested in them. What can we do with natural numbers? Well the basic axioms allow only one thing: Increment them.

Now, it is a natural question to ask what happens when we increment repeatedly. This leads to addition of natural numbers. The next question is to ask is whether we can undo addition. This leads to subtraction. Next, we ask whether all natural numbers can be subtracted. The answer is no. Can we extend the natural numbers such that this is possible? Yes, and in come the integers.

Now, that we have addition. We can ask whether we can repeat it. This leads to multiplication with a natural number. Next, we ask whether we can undo it and get division and rational numbers. We can also ask whether multiplication makes sense when both operands are non-natural.

Now, that we have multiplication, we can ask whether we can repeat it. This gives us the raising to the power of a natural number. Can we undo this? This gives radicals. Can we take the root of any rational number? No, and in come rational field extensions including the complex numbers.

A different train of thought asks what we can do with mixing multiplication and addition. An infinite number of these operations seems strange, so let's just ask what happens when we have finite number. It turns out, no matter how you combine multiplication and addition, you can always rearrange them to get a polynomial. Formulated differently: Every branch-free and loop-free finite program is a polynomial (when disregarding numeric stability). This view as a program is what motivates the study of polynomials.

Now, that we have polynomials, we can ask whether we can undo them. This motivates looking at roots of polynomials.

Now, we have radicals and roots of polynomials. Both motivated independently. It is natural to ask whether both trains of thought lead to the same mathematical object. Galois theory answers this and says no.

This is a somewhat surprising result, because up to now, no matter in which order we asked the questions: Can we repeat? Can we undo? How to enable undo by extension? We always ended up with the same mathematical object. Here this is not the case. This is why the result of Galois theory is so surprising to some.

Slightly off-topic but equally interesting is the question about what happens when we allow loops in our programs with multiplication and addition? i.e. we ask what happens when we mix an infinite number of addition and multiplication. Well, this is somewhat harder to formalize but a natural way to look at it is to say that we have some variable, in the programming sense, that we track in each loop iteration. The values that this variable takes forms a sequence. Now, the question is what will this variable end up being when we iterate very often. This leads to the concept of limit of a sequence.

Sidenote: You can look at the usual mathematical limit notation as a program. The limit sign is the while-condition of the loop and the part that describes the sequence is the body of the loop.

Now that we have limits and rational numbers, we can ask how to extend the rational numbers such that every rational sequence has a limit. This gives us the real numbers.

Now we can ask the question of undoing the limit operation. Here the question is what undoing here actually means. One way to look at it is whether you can find for every limit, i.e., every real number, a multiply-add-loop-program that describes the sequence whose limit was taken. The answer turns out to be no. There is a countable infinite number of programs but uncountably infinite many real numbers. There are way more real numbers than programs. In my opinion this is a way stranger result than that of Galois theory. It turns out, that nearly no real number can be described by a program, or even more generally any textual description. For this reason, in my opinion, real numbers are the strangest construct in all of mathematics.

I hope you found my rambling interesting. I just love to talk about this sort of stuff. :)