\ has no specific CSS meaning aside of escaping the next character. Here it allows to include the : as part of the group-hover:block class name and avoids interpreting it as a pseudo-selector (all of which are starting with the : character).
Regarding the hover control, it is actually implemented through the inline styles of your linked example. For instance <a class="font-semibold whitespace-no-wrap text-blue-600 hover:text-blue-800" means "apply the font-semibold, whitespace-no-wrap and text-blue-600 CSS classes on this anchor, and apply the text-blue-800 class on it when it is hovered".