Discovery Story Quests in Final Fantasy VII Remake are hidden scenarios you can unlock by completing various tasks. They appear as Purple Story Quests under the Story Main Menu.
You do not need to complete this Discovery Story Quests in order to finish the game. They are merely hidden secrets that players get rewarded with when they complete various tasks.
Chapter Three - Alone At Last
In order to unlock the Alone At Last discovery quest, you must complete all six side quests in Sector 7 Slums. This can be done as early as Chapter 3.
Once completed, Cloud and Tifa will head back to Tifa's apartment at Stargazer Heights where you'll speak to Marle, who gives Cloud a lecture. You'll also be asked to change out the Water Filters in your room.
Once done, head to Tifa's apartment. Tifa wants to know more about what happened to Cloud when he left the Village.
Eventually, Tifa suggests you to dress-up and go on the town. This is where you get to suggest an outfit for Tifa.
Chapter Six - Collapsed Passageway
In Chapter Six, there is a piece of Materia you can get to towards the end of the level. Once you turn off two Sunlamps and reach Treasure Chest (7) you can head down a ladder to the Lower Level Plate.
As you head down to the Lower Level Plate, you'll come across two Queen Grastrike mobs. They're weak to Ice Materia, so when you have your ATB charge, send some their way.
Once dispatched, you can use the power console to control the Cat Walk bridge. Line it up so you can cross to the next section. If you move the bridge as low as possible, you can also use it to pick up MP UP Materia (12).
Cross over and you can pick up the Orb of Elemental Materia (8).
Chapter Six - Inside The Ventilation Fan
Once you have deactivated all three Sunlamps in the Sector 4 Plate area during Chapter Six, you can use a moving platform to take you towards the Extractor Fans.
It will take you to a platform where you can climb up a ladder to a small room. There is a Vending Machine where you can purchase a Music Disc - #18 Electric de Chobobo.
Here you'll have to complete a task: take down a Queen Grashtrike and two Sentry Ray Gun within 60 seconds. It's pretty easy- just make sure you deactivate the timer once completed. In the same room, you can pick up a Hi Potion from a Treasure Chest (10).
You can now move behind the Extractor Fan and pick up the Summoning Materia: Chocobo & Moogle Materia.
Chapter Seven - Waste Recovery
In Chapter Seven, after you unlock the security door in B1, you can also access the Disposal Room and find rewards for disposing of any M-Units.
You will have to complete a mini-game, which requires you to pull the levers at the same time as Tifa and Barret. This takes about four different rounds to unlock the room.
Inside you can find Magic Up Materia on the floor and open any M-Unit capsules you have discarded.
You can find AI Programming Cores which sell for 500 Gil each.
Chapter Eight - The Gate Won't Open
When you take the backstreets with Aerith in the Sector 5 Slums, you'll come across a metal gate that seemingly won't open.
You can open this gate by following the path all the way round; you'll come to an opening area where a Smogger spawns.
On the wall is a ladder you can climb up and then open the gate. Fun!
Chapter Eight - The Language of Flowers
Aerith will ask you to follow her to some of her flowers. You'll get a small insight into who Aerith is, which ties into the bigger plot of the whole Final Fantasy VII Remake Story.
Chapter Nine - Vagabond Johnny
When you first arrive at Wall Street during Chapter Nine, Johnny will run past you as you head to Don Corneo's mansion activating the quest.
Completing the quest is easy; just find Johnny outside the Honeybee Inn and talk to him.
Chapter Fourteen - Johnny's Lost Wallet
Complete the quest Tomboy Bandit and retrieve Johnny's lost wallet.
Then head to the Sector 5 Train Station where you can find Johnny and give him his wallet back.