WiiConnect24 meant that your console would always be online, even when you weren't gaming. The Wii would basically "sleep" instead of being turned off completely, and could download content while not in use (think firmware updates, news or weather reports, messages from your friends, or Animal Crossing downloads). That way gamers would have up-to-date information available when turning on the console without having to manually connect to the internet. Wii software would just process whatever downloaded data was already available instead of having to connect and download on startup.
The problem is very few games ever made use of this feature. Few games ever ventured online, so having a way to download or update in-game content when gamers weren't playing really wasn't an issue. So a neat idea that really didn't turn into very much... why bother making it a selling point for Wii U?