Meet the condition to unlock the burden. Burdens can be enabled/disabled from the Burdens & Blessing menu when starting a new game.

Burden	How to unlock
Breakneck Burden	Stun enemies 50 times.
Deadline Burden	Complete a level 20 times before the time limit.
Deadly Burden	Take 100 damage from enemies.
Eclipse Burden	Experience 12 night time levels.
Excessive Burden	Die 10 times to traps.
Rebirth Burden	Kill 50 total enemies.
Stout Burden	Deal 500 damage to enemies.
Unhelpful Burden	Use genies 50 times.