The more Dream Points that are accumulated in a single play session, the happier the dreaming Pokémon will become. This is reflected in the icons which appear above the Pokémon when it is hovered over with the mouse.

Source
If you manage to accumulate over 500 Dream Points in one dream state of the Pokémon, it will raise a level upon your next Game Sync.- Serebii
According to this link that means every time you gain over 500 dream points before your Pokemon is sent back to the game card, your Pokemon levels up.