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It means that the horse has pretty much stopped adding points to its show score with training. It is still a good idea to keep on training them, in most cases.

In practice, most horses do continue to add minute fractions of a point with each training, but it can take months and months to increase the score by a point. If a horse levels off early at the very top of a Level/Grade, it may, eventually, bump up into the next higher Level/Grade, going from the top of its classes to the bottom.

If it's in the middle of the class score range, it may stay there for the rest of its life, slowly (at less than a snail's pace) creeping up toward the top of the class, but never getting out of it.

Before Ammit introduced the Showing Bonus, a horse that Leveled Off in a bad place was not really worth keeping for showing. Now, however, if it has a decent number of points, it should be kept (providing you have barn room) because it will still add to your stable's income until it passes out of the game from old age.

Thank you so much for all of this information! It's very interesting and you explain it so well.