The Mana Pool in Dragalia Lost can simply be viewed as a unique rendition of a "Skill Tree" of the Adventurers you have in the game. It is practically where you'd passively upgrade the stat of your adventurer (mainly STR and HP). Its is also where you unlock / learn passive skills and attack skills of that pertaining character. While you generally improve your character's basic stats, this in return increase its overall battle effectivity.
Fields to Improve include:
MAX HP node - Increases effective Hitpoints of your character
MAX STR node - increases effective strength (relative to attack) of your character
Skill Node - Unlocks either a passive or attack skill
Level Up Node - Unlocks the next tier of Mana Circle for that pertaining adventurer (usually becomes available after unlocking all nodes on that tier)
Each node would usually require a certain amount of mana along to actually upgrade. If you happen to not have enough mana to push through with an upgrade, you can freely re-do the given chapter / or event a number of times to earn as much mana as you want. However, unlocking the concluding tier of the Mana Circle also needs special items along with the mana points. The required item is different for each adventurer, but you'll most likely earn them as loots from doing the Endeavors in the game.