The axis of our planet is tilted about a third of the way over — 23.4 out of 90 degrees from what ought to be a straight up and down 0 out of 90 degrees. Through the miracle of planetary astrophysics, this axis tilt points in the same direction as the planet travels its annual circuit around the Sun. This means that at either end of the circuit the upper or lower half of the globe either leans in closer or further away from the sun to catch more or fewer warming solar rays on either its northern or southern ends. In effect, summer and winter switch back and forth between the two hemispheres. To answer the question: fall and spring happen in the in-between times on the annual circuit.
In theory, the root-cause of the tilt was the collision of one or more planetesimals, with Earth and the other planets as they accreted during the formation of the Solar System. Like braking billiard balls, each of the eight or nine planets was left spinning and wobbling around the Sun with varying amounts of degrees of tilt. Mercury is almost zero, Venus – 2.6, Mars – 25, Jupiter – 3, Saturn – 27, Uranus – 82, (which is nearly sideways) Neptune - 28 and Pluto – 57 degrees.
Other planetary motions include gravitational drag by moons and tides. For instance, the Moon no longer spins because of its closeness to Earth and neither does Mercury because its too close to the Sun. The ocean’s tides caused on this planet by the moon are slowing it down through friction. And then there’s the tendency for almost everything in the Solar System to get old and tired, fall into the Sun to burn up.
One of the great benefits of the seasons is they give Earth an even, annual toasting, otherwise heat would be concentrated at the equator and denied altogether at the poles. This, and the fact that we travel around and around the Sun at a certain, steady, safe distance, allows life to happen here in the Goldilocks Zone where the planet’s temperature is not too hot and not too cold but just right.