FFT's were very difficult for me to understand (and still are). The graphic at the top is actually a very good way to show what a practical FFT does. I wish I had it back when I first started trying to understand them.
There is an intuition to it, something like this: suppose we want to calculate a 2048 element DFT. If we instead calculate a pair of 1024 element DFT's over halves of data, we have all the same high frequency information there in the two windows. What we're missing is the lower frequency: the one that goes through just one cycle over the 2048 window. But we don't need 2048 points for that; the low frequency doesn't contain that much information. The FFT reveals how it can be obtained from the two halves. The two halves have the necessary information in their lowest frequency component; we don't need to sample 2048 points of the signal; we just look into the half-sized FFT results and put that together.
There are two reasons I like this approach. The first is that it is the only way to understand the number-theoretic transform, which is very useful in cryptography (among other things). The second is that you can use the same reasoning to divide the DFT in other ways (e.g. into thirds, or fifths, etc.). The downside is that it is more abstract so if all you care about is signal processing you need to somehow connect abstract "roots of unity" with complex frequencies (but that is probably not too bad if you are comfortable with complex frequencies).
That machine is incredible.
I can't begin to imagine the amount of work it took to design and buuld sonething like that.
Also, the video is very well produced.