I just took a semester-long seminar on the analysis of highly-chromatic 19th century music - it was extremely interesting (and intense... whew!), and we spent an entire week on "Tristan."
"Tristan" seemed shockingly chromatic at first since the public had not yet heard the Ring cycle, where Wagner developed many of his chromatic techniques. The "Tristan chord" has been hotly debated ever since the premiere, with many different theories on if the actual chord contains any non-harmonic tones or if it's just a half-diminished seventh chord.
The thing that makes the Tristan chord so slippery is the multitude of resolutions that sound nearly tonal to our ears, yet Wagner provides non-functional resolutions. As you said in your question, Wagner is known for planning out large-scale key relationships and this occurs in Tristan as well - the Tristan chord is often arpeggiated ("Liebestraum") and occurs in key areas of certain sections [F -> B -> D# -> A]. The most interesting aspect of this chord is the complete neutrality of the sound - it could resolve to nearly any other chord and sound "correct." I believe this is why Wagner was obsessed with it while writing "Tristan" and praised one of the early analyses for understanding exactly his writing process.
As for "laymans terms" this is pretty difficult without explaining Neo-Riemannian theory. Ordinarily we could analyze tonal music with Roman numerals, even taking into account chromatic inflections of secondary dominants or borrowed chords. Wagner's music is non-functional in that not every chord relates clearly to the key area - if there is one. Neo-Riemannian analysis provides us with a way to analyze triadic progressions with three basic transformations - P, L, and R - to get from one chord to the next.
For example, the Tarnhelm motive [G#m - Em] is fairly easy to analyze in Roman numerals (G#m: i - bvi) but G#m is not expanded on as a tonal tonic in the section. It makes more sense (once you know the N.R. analytical process, that is) to label the Tarnhelm progression as LR. Instead of relating triads back to a tonal center, Neo-Riemannian analysis looks at how the composer gets from one triad to the next in the least amount of transformations.
To sum it up without lecturing on PLR transformations, Wagner's chromaticism relies on smooth voice leading between triads and deceptive resolutions of expected melodic lines. Take the kiss motive from "Parsifal" - simply a chromatic scale moving through all twelve pitches. Instead of harmonizing this chromatic scale in the traditional way (I-V-I or some version of this progression with secondary dominants), Wagner inserts other non-functional resolutions of each pitch. Where we could not make sense of these triad movements in relation to a key center, Neo-Riemannian analysis would simply reduce them down to 1-3 transformations.
EDIT - forgot to add two incredible resources on the analysis of chromatic tonal music, including many Wagner examples:
Daniel Harrison - Harmonic Function in Chromatic Music
Richard Cohn - many articles in the Journal of Music Theory