Final Fantasy VII Remake has 60 Manuscript Locations. The Manuscripts earn you 10 SP each which is used to upgrade your weapon level and unlock new skills.

Weapon Level 6 is the maximum. To see your weapon level, open the Pause Menu > Upgrade Weapons and it will show the Weapon Level for each character and how many SP you need to the next level. The Weapon Level affects how many Sub-Cores you have available to choose weapon skills from (Pause Menu > Upgrade Weapons > Select a Weapon). A higher weapon level lets you select new skills, for example additional Materia slots or "Reprieve" which makes you survive deathblows with 1HP. These higher-level skills can be very helpful on hard difficulty. Weapon Level 6 requires 250 SP but on the first playthrough you can only get 246/250 SP at most, meaning you must play hard difficulty and get one more Manuscript per Character to get to Weapon Level 6. After this the additional SP are used to buy more upgrades for each weapon.

The vast majority of Manuscripts is only obtainable on hard difficulty, which unlocks after beating the game for the first time. After your first playthrough you unlock Chapter Select and it shows how many Manuscripts you got left in each Chapter (Pause Menu > System > Chapter Selection > check book icon on each Chapter). You can also check your collected Manuscripts under Pause Menu > Inventory > Manuscripts.

Simply put: All Manuscripts are automatic rewards from defeating all bosses on hard and doing all side quests on hard. A few are from doing the Colosseum Challenges (Chapter 9 & 14), Combat Simulator Challenges (Chapter 16) and some are bought from the Moogle Merchant in Chapter 8. One Manuscript replaces the Chocobo & Moogle summoning Materia in Chapter 6 (the only collectable one, everything else is a reward and not picked up anywhere). Just do all of this and you'll get all Manuscripts without going out of your way.