When you look in-game at the Fighter Board (in the Felyne kitchen at the right), you can see different options to select. Select the first option, "Development". When you select a Felyne you can choose different options. The options are as follows.

Fencing: The Felyne gains both attack power and defense power.
Martial Arts: The Felyne gains levels like in missions.
Meditation: The Felyne gains skill points.
Rest: The Felyne increases its loyalty.
Strength: The Felyne gains attack power.
Toughness: The Felyne gains defense power.