"Can someone please explain to me how "resolution inside a resolution" works?"
The terminology is confusing and misleading.
The correct explanation is "resolution using non standard, user defined timings"
The idea is to take a given resolution, example, 856x480, and lock it to a user defined set of timings (scan rate, refresh rate, dotclock value) normally used by another resolution....in this case of this guide, 960x540.
1080i HDTVs want to receive a scan rate of 33.75KHz and a refresh rate of either 30Hz (interlaced) or 60Hz (progressive scan).
So, in the case of all the custom timings and resolutions listed in the first post of this guide, ALL of them are locked to these HDTV safe timings using Powerstrip."For instance, can you set your computer output to display 800x600 and it is telling the TV to display 640x480p?"
Yes. Windows will set the screen resolution to 800x600 but Powerstrip will set the timings normally used by 1920x1080i so the HDTV receives compatible timings.
For a 1080i HDTV, any resolution above 540p must be interlaced as it can display either 1080 interlaced vertical lines or 540 progressive scan lines.
Therefore using the example above, 800x600 would be interlaced, and windowboxed in the center of the screen...pretty ugly, by the way. Interlaced resolutions should be used for media playback like DVD or gaming, use the progressive scan rez for desktop viewing of text...much clearer.
Everything you need to learn to get it going is in the first post of this thread. Good luck!