Weight can vary between -10kg and +10kg compared to a character's normal weight.
You gain weight by waiting in battle and eating food (=health items).
Also sometimes in your home base you are offered one of several meals, each with a + or - value. Choosing meals with a + before the number increases weight.
Your goal is to have any character reach a weight of +10kg.
To get this trophy pretty quickly play any dungeon until you reach the save point and then simply do a few easy battles with your character of choice waiting or eating food all the time and the others doing the fighting.
Then return to the save point, save and revive the enemies, then repeat.
If you focus on the one character being inactive and eating food you should reach -+10kg pretty quickly.