The problems the patch fixes were nothing phenomenally game breaking if you only ever played the game offline - but if you were one of the many that took part in races over the internet, it quickly became obvious that many people had discovered unfair, glitchy shortcuts that they could use to gain an unfair advantage – and it's these that the update removes. Currently, three tracks are affected by the patch – both the WuHu Island themed tracks, (WuHu Mountain Loop and WuHu Island Loop), where driving off the sides of the tracks at certain points caused you to be put back on the track much further ahead of where you fell off, as well as GBA Bowser Castle 1. Chances are, if you didn't know the glitches existed, the only thing you'll notice now are much fairer races online.
In order to download the update, you'll need to head over to the eShop, where you should see the update data on the front page, next to the 'Nintendo 3DS Download Software' icon on the bottom row. If you can't see it there, you could always use the search bar at the top of the page, where searching for 'Mario Kart 7' should take you to the page for the game, where you'll be able to find the Version 1.1 Update Data. Before you can access the eShop, you may need to update your system – the act of accessing the eShop should prompt it to ask, but if not, you can always head into the Settings section of your Home menu, followed by 'Other Settings', then, on the final page of that screen, you should see 'System Update' – click this and away you go.
You might be wondering – why would someone bother going all the way to the eShop to download an update that effectively removes their easy route to first place? Sensibly enough, Nintendo have made it impossible to play online races unless you have the latest update, so the cheating scumbags will be forced to patch their game if they ever want to play online again – which is good news for the rest of us, really. Even if you don't download the update though, you can still play the Single Player and Local Multiplayer.