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A common misconception is that the Earth is closer to the Sun in summer, but it's actually further away. Diagrams would be better for explaining this. I'll get around to it, but I'll try to do it with words for now.
Less atmosphere to pass through
Because of the greater elevation of the Sun in summer, the distance light has to travel through the atmosphere is shorter than it would be if the Sun were low in the sky. The Sun feels hotter in summer because of this and also the filtering effect is the reason the Sun looks dimmer and redder when it's low in the sky near sunset as light gets scattered and absorbed. Because of the higher intensity of illumination of the Earth's surface in summer, this makes air temperatures higher as the ground warms up and transfers its heat to the ambient air. This atmospheric filtering effect wouldn't happen on the Moon, irrespective of the Sun's elevation, because there's no atmosphere.
Lower angle of incidence
Because the Sun is higher in the sky, when solar radiation hits the Earth's surface, it's spread out less than in winter (compare the difference between a torch shone perpendicular to a wall and at an acute angle). Greater solar irradiance, measured in watts per square metre (W/m²), warms the ground more, which in turn heats the air above it. This is why solar panels are angled so that the Sun's rays strike them as close as possible to right angles, maximising the irradiance they receive.
Longer days
Because of the tilt of the Earth, days are longer and nights are shorter. This gives more time for the Sun to warm up the ground, raising the average air temperature higher than in winter. Night-time temperatures remain mild because the ground acts as a thermal reservoir and stores heat. If the Sun mysteriously disappeared, ground and air temperatures would continue to fall over days and months to a level hundreds of degrees below 0 °C.
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