Why the temperature of the air in the atmosphere hotter in summer than in winter?
The ground is receiving more direct sunlight and is therefore hotter. This heat is released into the atmosphere.
To begin, you need to understand that the Earth's axis is on a tilt. When one part of the Earth is tilted toward the sun then the other half is tilted away from the sun.
During the summer months, the Earth is tilted so that the hemisphere in question (Northern or Southern) is tilted toward the sun. As a result the angle that the sun's energy hits the Earth at is more direct, and therefore more energy is absorbed by the Earth.
The atmosphere is not heated by the sun however, but by the Earth. The warmed Earth releases energy in the form of long wave radiation (heat) and this warms the atmosphere.