A hard keeper is a horse that requires more, like a TB for instance are generally harder keepers cause they can't keep weight on very easy and just require more attention.
A easy keeper is one lets say that gets fat just on hay, doesn't require special attention they just take a lot less to care for.
An easy keeee usually refers to a horse that is easy to maintain its body weight. It doesn't gain or loose weight too quickly. A hard keeper would be the opposite: difficult to maintain a healthy weight. It usually refers to a horse that looses weight easily, although it could also apply to a horse that gains weight too quickly.