When you look in-game at the Fighter Board (in the Felyne kitchen at the right), there are different options to select. Choose the first option, "Development". When you select a Felyne you can choose the following options:

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