https://nandgame.com/ will guide you through using NAND gates to create one computer. (There's lots more to learn about how to create other kinds of computers, and other ways of designing them, but walking through that one example should at least dispel the illusion of magic.)

The last time I did it, it took me two hours, but I think that if you are coming at it without any previous knowledge of digital logic it should be tractable within about 40 hours.

Then the only remaining things to understand are:

① how can you create NAND gates with transistors?

② why are transistors better for this than, for example, relays, pneumatic valves, or neon tubes, all of which would also clearly work?

The answer to ① is actually pretty simple; the best explanation I've found is in The Art of Electronics, though Ken Shirriff's blog has a lot of great stuff on it too. And Falstad's circuit.js has some good simulations, like https://tinyurl.com/2eegeyxj and https://tinyurl.com/ybpo5ls2, which I think are nicely complementary to the theory.

I've never seen a good explanation of ②, but I think the main answer is that electrons are about ten thousand times lighter than atoms, so with the same energy, they move about 100 times as fast. The result of that is that switches that work by moving around electrons can switch a great deal faster than switches that work by moving around atoms. Neon tubes also work by moving around electrons, but they're slower than transistors for a different reason: it takes the gas a long time to thoroughly deionize.