I still have no fucking idea what Spirits is.
SOMEONE PLEASE EXPLAIN
So there are at least two modes that feature Spirits: the story mode, World of Light, and this game's version of event mode, called Spirit Board.
Spirits consist of videogame characters — hundreds of them, probably thousands — that have been separated from their bodies by the events at the start of World of Light.
They are concept art/renders wreathed in light. And they come in two flavors: Primary Spirits, and Support Spirits.
Primary Spirits have varying power levels. Your fighter can equip one at a time, the Primary Spirit adding its power to your fighter.
Primary Spirits also have varying numbers of slots. In these slots, you can add Support Spirits.
Support Spirits don't increase your power. However, they do bestow modifiers. They can make you faster, larger, invisible; they can grant you super armor, make you jump higher, grant you resistance to wind or poison or weapons; they can make you spawn with certain items, automatically heal you or fill up your Final Smash meter faster.
The main way you collect Spirits is in Spirit Matches. Again, these appear in both Spirit Board and World of Light. Here, you fight an opponent — a Puppet Fighter — who has been possessed by a Spirit. And that Spirit has altered them.
For example, in one event, you fight a Puppet Fighter of King K. Rool who has been possessed by the Spirit of The Imprisoned from Skyward Sword. This has changed King K. Rool so that he is supersized, and unflinching, and black in color; he stands perfectly still in Gerudo Valley, and you must chip away at his stamina to defeat him; but the Spirit has also spawned an Assist Trophy of Ghirahim that constantly attacks you.
In other words, it recreated The Imprisoned's scenario from his base game, Skyward Sword. And upon winning, you'll have a chance to collect the Spirit. A shield will rotate around the defeated Puppet Fighter, and you have a limited number of shots from a special gun to hit the Puppet Fighter by shooting through gaps in the rotating shield. If you hit the shield, part of the shield will disappear so that it's easier to collect the Spirit next time.
Spirits can be leveled up by being used in battle or fed Snacks. This makes them stronger. Upon maxing out their level, some Spirits will evolve into stronger forms — for example, leveling up the Dixie Kong Spirit will evolve it into a Spirit of Dixie Kong and Kiddy Kong in the hovercraft from DKC3.
You can also buy Spirits with gold in the in-game shop. Sometimes you'll get duplicate Spirits. You can reduce your duplicate Spirits to their cores — there are three types: Attack, Shield, Grab — and then you can combine these cores to craft all-new Spirits.
Spirits can also train with Spirit Masters to learn new skills, modify their properties, grow stronger, etc, or even be sent on expeditions to collect loot for you while you're off doing other things.
There's a lot more, but that's the gist of it!