I'm going to give a very cursory simplification for the answer because asking about Lydian Chromatic theory is just like asking about Set Theory or Serialism.
Lydian Chromatic Concept Theory basically asserts that the lydian scale is more closely aligned to the natural, universal properties of sound than the conventional major scale. It explains and justifies this reasoning through the overtone series, interval vectors, and what is described as tonal gravity. Tonal gravity is the aural relationship between a given note and the fundamental of it's prime-order lydian chromatic parent scale. A tonal gravity field is something not unlike a pitch-class region.
Because the perfect fifth is the first interval introduced after the perfect octave, it is
"thus established as the strongest harmonic interval" (pg.2).
In other words, the fifth is considered to be the "foundation" or "cornerstone" interval. To this end, the fifth then establishes itself as the basic unit of tonal gravity whereby
"a ladder of fifths proceeding upwards from the tonic...produces the
first seven tones of the Lydian Scale..." (pg.3).
When discussing this theory you need to dispense with functional harmony. That said, this theory may be used to create tonal, atonal, and pantonal music. It is important not to conflate functionality with tonality. Tonal Gravity is measured vertically as well as horizontally.
Here is another quote from the text in question that might help:
There is no "goal pressure" within the tonal gravity field of the Lydian Scale. The Lydian Scale exists as a self-organized Unity in relation to its tonic tone and tonic major chord. [It] implies an evolution to higher levels of tonal organization. [It] is the true scale of tonal unity and the scale which clearly represents the phenomenon of tonal gravity itself.
"Goal Pressure" here of course refers to the depends on the Dominant -> Tonic relationship. "Unity" is the process by which "gravitational energy is passed down a ladder of fifths to its lowermost tone..."
Each Chromatic Order has seven principle scales. These principle scales (along with most things in this theory) are derived from the overtone series.
From there on, it just gets a lot more complicated.
The Take Away
Lydian Scale is more closely aligned to the natural, universal properties of sound than the conventional major scale.
The harmonic series is the most natural expression of sound at its fundament.
All musical grammar and expression of this language evolves from the harmonic series.