The trick on https://intertwingledinnovations.com is that some of the colours are specified using the display-P3 colour space (also known as “wide colour” or HDR). This is a relatively new standard for expressing a wider range of colours than the familiar sRGB colour space. This diagram from Wikipedia shows the subset of the range of all possible colours that are supported in different colour spaces. You can see how P3 covers colours that are out of reach of sRGB:
Not all devices/displays support P3, but it is getting more common, particularly on phones and tablets. Even for those that do, colour pickers in applications tend to be restricted to sRGB colours.
The upshot of all of this is that there is a set of secret colours that are very striking when you come across them. I find that pretty awesome. P3 is typically used in small doses, for example brightening up sunsets in photos, but it amused me to experiment with using it in a more unexpected way. My hope was that visitors would be struck by the vibrancy, and be intrigued.
However, I take the point about inducing eye strain, and if that is a widespread problem I would be happy to change it.