How does being underleveled affect battles?

The penalty starts early: being one or two levels below an enemy is enough to trigger it. At that gap the enemy already gains damage multipliers and resistances against you, so you do not need to be badly outclassed before a fight turns against you.

The level that matters is your Aniimo's. Bring an underleveled one into a fight and both sides of the exchange get worse together. Your hits land for noticeably less and theirs land for noticeably more. You need more time to win and can take fewer mistakes while you do it.

What still helps when you are behind

Fight as the creature. In trainer form you cannot dodge, but when you twine with your Aniimo you can dodge attacks yourself, and that keeps it alive longer than the AI manages. When every hit taken costs more, avoiding hits is worth the most.

Work the Break gauge. Depleting an enemy's break meter staggers it, lowers its defenses and opens a window for burst damage and Ultimate attacks. Save your strongest skills for that window, since your normal damage is being cut by the level gap.

The largest bosses are designed to be fought in groups, so a co-op party is the other answer to a fight you are underleveled for.

Catching higher level Aniimo

Capture works by weakening a wild Aniimo in battle before throwing an Anipod. Against a higher level target your reduced damage makes that weakening slower, and you take more damage while you do it.

Not settled: The exact size of the multipliers and resistances is not published, and neither is how much worse they get once the gap grows past two levels.