The bottom line for the changes from season to season is the average daytime temperature. This depends on the amount of heating that the earth receives in a single day throughout the year, and this depends on how many hours the Sun is above the horizon and exactly how long it spends at its highest elevation above the horizon. For every square meter on the surface of the earth, it will be heated by the Sun at a rate that depends on the 'cosine' of the angle of the Sun above the horizon. The higher the Sun gets, the less slanted the rays of light are that intercept each square meter, and so the efficiency with which these slanted rays can deliver energy to the surface gets better and better the higher up the Sun gets. When you add up during the daylight hours just how much heating this surface gets, it receives most of its heating from those times during the day when the Sun is the highest above the horizon. For a tilted earth, there will be some days during the year at a given latitude, where this heating rate is the highest and we call this summer. There will be other days when the Sun never gets very high above the horizon and so its heating ability is very low, and we call this winter. The details of just how hot and cold we get, and the exact dates, depend also on whether we are near water, or in the interior of a continent.
So, seasonal changes depend on the tilt of the earth's axis because they lead to changes in the amount of heat delivered to a square meter of surface, and the fact that there are a changing number of hours in the day when the Sun is above the horizon and high enough up that it can efficiently heat the surface over the course of a typical day.