The Training Area is a platform of the Battle Tower. It is staffed by either Hermosa or Kenneth depending on who you chose for the quest Sparring Partner.
By speaking to your trainer, it allows you to purchase an increase of the max potential of skills.
Base Platform
The initial platform has your trainer as well as a locked chest & a Dragon Skill Book.
First Upgrade
After completing the first quest, Sassan will allow you to upgrade your platform which increases the level caps allowed.
Second Upgrade
After completing the second quest, Sassan will allow you to upgrade your platform again with increased level caps.