Hex, RGB and HSL are three dialects of one sRGB triple
A color on a screen is three channels - red, green and blue - each a byte from 0 to 255. Hex writes those bytes as two hexadecimal digits each; rgb() writes them decimally; hsl() re-describes the same triple as hue (the angle on a color wheel), saturation and lightness.
The conversions between them are exact within sRGB, which is what the page’s "same pixel" claim rests on. A wider gamut or a printer needs a device profile - a different space entirely - and the page deliberately does not fake that conversion.
Automatic mode detects the format from the text - a leading # points to hex, rgb( to rgb and hsl( to hsl - and falls back in that order, so pasting any of the three usually needs no mode selection at all. The detected format is printed beside the result, so you can see how the page read your input.