Hero's Trace is a solo instance where you complete stages which have a series of challenges. The more challenges you complete, the more cursed magic crystals you get which are used to purchase a variety of rewards.
Once you reach base leve 150 Hero's Trace's are unlocked in the Instance finder.
There are currently 12 stages with each stage having three challenges like the ones in the image above. These challenges range from simply clearing the stage, clearing very quickly, and avoiding certain attacks to name a few. You'll want to clear each stage while completing as many challenges as possible to earn more Cursed Magical Crystals.
You can run this two times per day to free, but need to spend 60 minutes of combat time if you want to do more. I recommend you only run this twice per day since there are better places to spend your combat time.
After clearing a stage for the first time you can use the raid feature to auto complete the stage in the future so clear as far as you can and use the raid feature to slowly earn more Cursed Magical Crystals.
Location of the "Raid" feature to auto complete an instance
To access the rewards shop open the instance finder and select Hero's Trace. A shop button is located at the bottom right of this screen next to a count of Cursed Magic Crystals available to spend.
Location of the rewards Shop
Several headwear can be unlocked from this shop. Some have decent stats for people who don't have great headwear yet and all provide deposits for your adventure handbook. You can also purchase some pink gacha slips, mastela Fruit, runes, and enchant materials. All items in this shop have a purchase limit so once you've cleaned out the shop the content is complete.
Examples of Rewards