Why should I keep my characters at low levels in Final Fantasy VI?

The only reason to keep your heroes low level until the end of the game would be espers: specific espers grant stat bonuses on level up, and the best ones are late game. This is by no means necessary and I don't recommend doing it unless you're really trying to min-max.

You should, however, attempt to have most characters at a decent level by the end of the game (40ish), as you'll need to use 12 characters for the final dungeon.

Note that characters do not gain stats on their own, they only game them from espers at level up.

For a fair chunk of the FF games (VIII, IX and IV), leveling up doesn't do much on its own. The only way to grow your characters stats in those games was via Espers (VI), Junctions (VIII) and Equipment (IX) that were equipped when the character levels. (The remake of IV also had characters gain stats based on what abilities they had equipped between levels 70 and 99, but to get "ideal" stats you needed to beat the game twice before, so that's clearly optional)

That means, for a certain subset of players, gaining a level without having the optimum Esper/junctions/equipment is a waste. If you wish to play optimally (in the case of 6, you can beat the last boss is one round before it has a chance to do anything), keep your levels as low as possible. It's by no means a requirement, the game is perfectly beatable without playing optimally. There are no extra scenes or achievements for playing the games "perfectly," it's personal preference.

(There are two other ways to get stats in FFVIII (other than junctioning) that have nothing to do with levels; the Devour ability and consumables that increase your stats)