Before you can tame animals you have to do a story based mission early on in the game. You will also need a total of 25 meat to unlock this, you can harvest meat from dead animals and often you will get some from your in game daily loot sack drops. It dosen't require the same tamed beast, you can heal 25 individual ones if you wanted to. Chances are you will get this whilst playing the game normally. But if you want the achievement quickly, all you have to do is deal some damage to your tamed beast and then heal him, rinse and repeat.
Just a side note, if you tamed beast dies you are able to revive him in the beasts menu at the cost of some herbs and meat.