The Library shows every Hero you've collected while playing Dragon's Watch.
Heroes are displayed in Hero Sets, and will display here even if you have fused them away so that they are no longer available on the Heroes screen. Completing an entire Hero Set will earn you a reward.
One Hero Set for each of the game's Regions has a slot for a Golden Guardian. To win the Golden Guardian, you must first complete that Hero Set. This is done by collecting the Hero for each of the slots in that Set. Generally, you'd do this by summoning/winning the Heroes on the left hand side of the Set, then levelling those Heroes up until they can be evolved (filling in the middle Heroes of the Set), then levelling/evolving them again, to reveal the right-hand Heroes of the set.
VariantsSome Sets in the Library have two separate, overlapping Hero Slots for each position. These are for Variants. Variants are Heroes which are similar (not not identical) in appearance, skills and strengths. You don't have to collect all Variants in a Hero Set to win the Golden Guardian - just one of the two Slots for each position will suffice.
The Dragon Egg icon in the bottom right of the Library screen shows you which Dragons you have rescued - one for each region. As of December 2017, the Dragons are not involved in questing or PVP. But...