I learnt this subject from the book Galois Theory, 5th ed. by Ian Stewart. Quoting from page 177:

Theorem 15.10. The polynomial t⁵ - 6t + 3 over ℚ is not soluble by radicals.

As you can see, this theorem occurs in Chapter 15. So it takes fourteen chapters before we reach here. It takes a fair amount of groundwork to reach the point where the insolubility of a specific quintic feels natural rather than mysterious.

To achieve this result, the book takes us through a fascinating journey involving field extensions, field homomorphisms, impossibility proofs for ruler and compass constructions, the Galois correspondence, etc. For me, the impossibility proofs were the most interesting sections of the book. Before reading the book, I had no idea how one could even formalise questions about what is achievable with a ruler and compass, let alone prove impossibility. Chapter 7 explains this beautifully and the algebraic framework that makes those proofs possible is very elegant.

By the time we reach the section about the insoluble quintic, two key results have been established:

Corollary 14.8. The symmetric group S_n is not soluble for n ≥ 5.

Theorem 15.8. Let f be a polynomial over a subfield K of ℂ. If f is soluble by radicals, then the Galois group of f over K is soluble.

The final step is then quite neat. We show that the Galois group of f = t⁵ - 6t + 3 over ℚ is S₅. Corollary 14.8 tells us S₅ is not soluble. By the contrapositive of Theorem 15.8, f is not soluble by radicals.

Obviously whatever I've written here compresses a huge volume of work into a short comment, so it cannot capture how fascinating this subject is and how all the pieces fit together. But I'll say that the book is absolutely wonderful and I would highly recommend it to anyone interested in the subject. The table of contents is available here if you want to take a look: https://books.google.co.uk/books?id=OjZ9EAAAQBAJ&pg=PT4

Two small warnings: The book contains a fair number of errors which can be confusing at times, though there are plenty of errata and clarifications available online. And unless you already have sufficient background in field homomorphisms and field extensions, it can take several months of your life before you reach the proof of the insoluble quintic.