1) Its the opening scene of the game. Rain and Lasswell are on a mission/patrol/whatever on an airship, when they both see a vision of a girl trapped in a crystal begging them to come to the
to prevent the
from being destroyed. Rain, being highly impulsive and easily persuaded by women, diverts from whatever the planned mission was to investigate.
There, they get their asses kicked by a typical JRPG-troped armored man in black who easily destroys the
Rain then decided that they need more power to get revenge for their loss by gaining powers from espers (and visions via Lasswell, because gacha). Esper dialogue is standard textbook "fight me to gain my power to assist you in saving the world" stuff that has existed in every FF since the dawn of time.
2) Espers require magicite to level up (magicite shards, magicite, megacite). They will only accept those types of items for leveling up. They have their own skill grid where you purchase abilities and stat bonuses for them (which carry over to the character that equips the esper) via stat points earned by leveling up. Grid selections are permanent, so be sure in the direction you want to go on the grid before investing skill points.