"Good-bye, darling." "Good-bye, sweetheart." Alix Martin stood leaning over the small rustic gate, watching the retreating figure of her husband, as he walked down the road in the direction of the village. 'Leaning over' is referring to Alix's physical position. She was in the garden, standing by the closed gate. Her feet and lower body were inside the garden, and she probably had her arms or hands resting on the top of the gate. Meanwhile, her upper body was inclining forwards, outside the garden, so that she could watch Gerald for as long as possible as he went down the road. The weight of her upper body was supported by the gate.

The basic meaning of "lean" is to stand at an angle, not straight up. In order to "lean" you usually need something to support you. You can lean against a post. You can lean against a car. She is actually leaning "over" a fence. Her feet are on one side. her hands and arms are on top of the fence. Her head is across the fence, on the other side. She is using the fence to support some her weight. This is a comfortable and relaxed position for talking to someone on the other side, or looking at whatever is on the other side. This is a picture of a boy leaning over a fence: http://tinyurl.com/h5j3aqb Figuratively, "leaning" can mean a tendency or a partly-made decision. "Should I order the chicken or the steak? I'm leaning toward the steak. Do you know this restaurant? Is the steak good here?"

It means bend over , tilt over