Once you defeat a boss, head to the teleporter back at the Cursed Outpost, and select the Breaker you want to unlock. Spend your Abyss Stones, and the selected character will be available to play.

To unfreeze your character, simply start mashing the jump button or equip a weapon from your inventory. To teleport to the portal cave, head back to the exact obelisk that acted as your teleporter ...