In this video, they explain that performing a chromatic inversion on Paganini's them changes the root note from Am to D#. Why does it changes both the root key and the mode, and how does that work? It seems that Am to D# is from the circle of fifths, but as to why I don know
D♭, not D♯.
But stick to the main part of the explanation, where the original melody, rooted in A and rising up a perfect 5th to E is inverted. It still ends up a 5th away from A, but this time a perfect 5th DOWN, ending on D, which now feels like the root. That's the end of story as far as the 'negative melody' is concerned.
Having got his new melody, Rachmaninoff decides to play it rooted on D♭. Not important. (Well, important in terms of orchestral sonority, and key relationship with the previous variation. But not important in how the melody was derived.)
Why does the lowest note become the root? Well, listen to it! The original outlines an A minor triad (plus one passing note). Unless context tells us otherwise, that's going to sound like the tonic triad of A minor. The inverted transformation outlines a D major triad. So the immediate impression is that we're in D major. In both cases the composers COULD have taken the music in a different direction (though neither of them did). But, in itself, a strongly stated triad at the beginning of a melody will be heard as a tonic.
I think you're skipping over something - he states that Rachmaninoff transposes the theme from the key of A minor to D♭ major. The key changes just because he transposes it - there's no way getting around that. I don't quite understand what you mean by changing the mode; I don't know either piece very well, but from what I can see there's no modes changing. Are you trying to ask why we're going from minor to major? Would love to know your question a bit better.
I can't comment on Laurence Payne's comment because of reputation, and I also think that the actual transposition/inversion order isn't really the crux of OP's question. At least we got OP's question properly:
What I fail to understand is that he still started from the same note, so why is it not same root? Why is the ending note is suddenly the root? Is it because it's lower than the starting note?
It's not because it's lower than the starting note. If you just consider the particular theme of
A C B A E to
A F# G A D, there really isn't any semblance to any typical A chord in A F# G A D. Most chords will have a fifth, i.e. an
E in this case. There's also no third, which isn't a biggie, but eliminates the possibility of
A F# G A D fitting into a major or minor chord.
If you're determined to fit
A F# G A D into an
A chord with
A as the root, you're looking at a chord with a major 6th, a minor 7th, and a perfect fourth on top of the root, which really doesn't make anything useful for our purposes. Simply put, you're not gonna fit this into a nice chord with
A as the root. The modified theme also doesn't really fit into the natural minor, harmonic minor nor melodic minor scales either. Raised 6th and unraised 7th is a little weird.
If you however consider this from the perspective of D being the root, you've got a pretty nice shape there.
D F# A is basically a D major chord. G is a perfect fourth above it, and it's going between two notes already in the D major triad; it works there.
So that's why I would think we changed roots and the variation is in a different key; rooting the theme over one chord with
A as a root isn't really suitable.
However, there's really no strict requirement saying that the theme had to be going to D major; going to
G major has a possibility of building around a
G9 chord, going to
B minor has a possibility of building around a
Bm7 if we just add a B underneath in the bass (and B minor is actually the relative minor of D major)
The point is, I would think the root got shifted because it's not easy to continue in A in this case. D works nicely in this case, but D wasn't the only choice. B minor and G are also choices, but D is easiest in this case, at least to me.
I'm not very experienced with music theory; maybe someone could care to double check this?