- a race or other contest in which certain disadvantages or advantages of weight, distance, time, etc., are placed upon competitors to equalize their chances of winning.
dicionary.com; there's more than one definition of the word handicap. That is its original meaning, rather than a derogatory word meaning a person with mental or physical disabilities.
In regards to why it's used:
>PSS battles let you set up a handicap before battling. If you set up a handicap, you can change the number of Pokémon participating in that battle. This is a way to level the field before battles between players of different skill levels.
This means the handicap is based on your ranking and (presumably but not confirmed) the Pokemon used.
Source (official Pokemon website)