Why is \LARGE 20% bigger than \Large and \Large 20% bigger than \large, etc.?
The LaTeX size names are related to the fonts available in the earliest releases and they in turn are related to this comment in the TeXBook:
\danger At many computer centers it has proved convenient to supply fonts at magnifications that grow in geometric ratios---something like equal-tempered tuning on a ^{piano}. The idea is to have all fonts available at their true size as well as at magnifications 1.2 and~1.44 (which is $1.2\times1.2$); perhaps also at magnification~1.728 ($=1.2\times1.2\times1.2$) and even higher. Then you can magnify an entire document by 1.2 or~1.44 and still stay within the set of available fonts. Plain \TeX\ provides the abbreviations ^|\magstep||0| for a scale factor of 1000, |\magstep1| for a scaled factor of 1200, |\magstep2| for 1440, and so on up to |\magstep5|.
To answer the extra question in the edit above
Note that smaller sizes like 7pt and 5pt are not (in computer modern) made by scaling down the 10pt font but are generated at that design size, so (more or less) the available fonts were base fonts at sizes 5pt, 7pt and 10pt, scaled up by magsteps of 0.5,1,2,3,4,5
The TeXbook describes this choice making a reference to the equal-tempering in musical instruments. Don't forget that Knuth is a musician himself and plays the organ.
The octave is divided in semitones having respective ratio the 12th root of 2. This gives slightly “untuned” notes, because, for instance, the dominant should have a ratio 3/2 with respect to the tonic, whereas
27/12 = 1.498...
(there are seven semitones to go to the dominant). The difference is very small, but noticeable for people with “absolute pitch”.
By choosing an “equal-tempered” scale based on 1.2, we have that the square root of 1.2 is 1.095 (not so different from 1.1) and scaling a 10pt font with these ratios we get
10pt 10.95pt 12pt 14.4pt 17.28pt 20.736pt 24.88pt
which are remarkably near to the point sizes actually used in metal typography:
10pt 11pt 12pt 14pt 18pt 20pt 24pt
see https://en.wikipedia.org/wiki/Traditional_point-size_names