Why does the day have more hours of light in summer time when you move further from the equator?

Answer

Actually, it is the other way around: because the days during our summer have more hours of light, it is warmer during the summer. Because the more hours of light, the more solar energy we receive, the warmer.
Why are there more hours of light now? As you say yourself, this has everything to do with the tilt of the axis of rotation relative to the Earth’s orbital plane around the Sun. This tilt is slightly more than 23°.
When summer starts here (solstice) the Sun is therefore perpendicular above the Tropic of Cancer (just over 23° north latitude). This means that the North Pole (northern axis of rotation) is illuminated by the Sun for 24 hours. So the further you move from the equator to the pole, the more hours of light there are in the day. The days at the equator always count the same number of hours of light, 12 hours. Hence, there are not really seasons at the equator.