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An ELI5 (summarized from the video linked below because I am not this clever or smart) is that each key represents a set of notes that are harmonious with each other. By changing keys, the set of harmonious notes changes. Keys that are close to each other share more harmonious notes than keys that are far apart, so the key of C and the key of G have very similar sets of harmonious notes except for one. The further away you get from our example of the key of C, the bigger the differences — there are still common notes, but fewer of them.
These common notes allow a musician to bridge from one key to another via these shared harmonious noted without the listener noticing, and then produce a dramatic and unexpected change. You were expecting something in the key of C, but the musician had switched keys via the common set of harmonious notes and you never noticed. They then introduced something that is harmonious to the key they were playing in, but unexpected because you were “listening in the key of C” and that’s drama!
The science is that the different harmonious notes are specific ratios of the base note that is being played. Others have explained those ratios better than I can. Look up what a chord is to get a sense of why those specific three notes played together sound harmonious and any other three notes doesn’t.
I can’t explain without writing a book. Instead, [I offer a video that explains it](https://youtu.be/62tIvfP9A2w), uses a few analogies, a visual model, and discusses John Coltrane’s Giant Steps to make sense of it. It doesn’t even get to keys until more than halfway through, with the first half being prelude explaining a bit of the structure of fifths. It’s a good explainer, but I had to watch it twice to help put it together myself.
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