There are 2 different things.
First, many games need min. specific version of firmware (for most of present games it is max. 4.5, only exception is Mario & Luigi: Dream Team Bros which needs 5.1) to run. If you have lower version, you will be asked to update before fist run.
Second, it is update of the game. Few games using that. Without latest update you are not able to play online. One is Mario Kart. Another will be Animal Crossing, but no update released yet. This update must be downloaded. To do that, you must be on the newest firmware. Always.
Btw. for Animal Crossing nothing will change if you update to 6.1 now. After update, when you will click this update button, you will receive msg that you have latest available version of game.