If you're looking for deeper explanations, I recommend "Practical Signal Processing": https://www.amazon.com/Practical-Signal-Processing-Mark-Owen...
Many of the chapters are here: https://battle.shawwn.com/sdb/books/PSP/ but buy the book if you can afford to.
In particular, Chapter 4's "untwisting" intuition is the best I've found: https://imgur.com/a/r7KvO60
Another way of phrasing it: You know that effect when a wheel is spinning so fast, it looks like it's going backwards? If it looks like it's standing still, there's a Fourier coefficient at that rotational frequency.
... Or at least, I think there is. It's hard to be precise with analogies.
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