How can you make Highlights of online races?
If you go to the Mario Kart TV option afterwards (from the main menu) you should have the record of the race in your "recent" section (from where you can edit & share). Though (I presume) it won't be there if you played enough in the meantime.
Worth noting that during the post-race cool down laps you can press the + button to star it, so that it does not disappear. You can have up to six starred replays, which can be viewed and managed in Mario Kart TV menu option.
Thanks for sharing videos everyone! These are really fun to watch. :)