Answer
Days are longer in the summer than in the winter because the Earth's axis of rotation is tilted at an angle of $23.5^{\circ}$ from the vertical as the Earth orbits the sun.
Work Step by Step
The Earth's axis of rotation is tilted at an angle of $23.5^{\circ}$ from the vertical as the Earth orbits the sun. For six months of the year, the north pole is tilted toward the sun, and for the other six months of the year, the south pole is tilted toward the sun.
For example, when the north pole is tilted toward the sun, people in the northern hemisphere can see the sun for more than a $180^{\circ}$ rotation of the Earth. These effects are stronger around the summer solstice which is the day of the year with the most hours of sunlight. Therefore days are longer in the summer than in the winter.
Note that when people say that "days are longer in the summer", the meaning is that we have more hours of sunlight in the summer. The actual length of a full day, that is the amount of time required for the Earth to rotate once, does not change.
