This is pretty straightforward. You probably haven’t missed the option in the menu that is used to turn on this mode. Hardcore is a third, separate mode like creative and survival mode. But it has a catch.
If you die, that’s it. Game over. You can’t respawn and you lose all your progress. The world you built is erased forever. What makes this even more challenging is the fact that you cannot change the difficulty in this mode mid-game. It’s always on the Hard setting.
Beating the Ender Dragon in this mode is extremely difficult. You have to make use of all the tools that Minecraft provides which include the potions, enchantment system and food buffs, and acquire the best armor in the game before you face the final boss.
You should enchant all your armor pieces with Protection V and Blast Protection as well. And when mining for resources to stay alive, keep in mind the golden rule — never dig straight down, lava is more common underground than you think.