What the display colour test shows
The greys are the diagnostic ones. A neutral grey should be neutral: if the 50% patch looks warm or blue, the panel or its profile has a cast, and that cast is affecting every image you look at without you noticing.
The primaries check each channel on its own. If red looks orange, or blue looks purple, something in the signal path is shifting it — a bad cable, an aggressive picture mode, or a colour profile the operating system is applying.
What to look for on each colour
On the greys, look for a tint and for uneven brightness across the panel. On the primaries, look for patchiness and for areas where the colour looks washed out or darker.
If you have two displays side by side, running this on both at once is the fastest way to see a cast that would be invisible on one screen alone — the eye adapts to a tint within seconds when there is nothing to compare against.
What this test cannot tell you
This is a viewing test, not a measurement. Judging colour accuracy properly needs a colorimeter held against the panel; a web page has no access to what the display actually emitted.