Eclipse seasons are the result of the axial parallelism of the Moon's orbital plane (tilted five degrees to the Earth's orbital plane), just as Earth's weather seasons are the result of the axial parallelism of Earth's tilted axis as it orbits around the Sun.
During the season, the "lunar nodes" – the line where the Moon's orbital plane intersects with the Earth's orbital plane – align with the Sun and Earth, such that a solar eclipse is formed during the new moon phase and a lunar eclipse is formed during the full moon phase.
During the eclipse season, the Moon is at a low ecliptic latitude (less than around 1.5° north or south), hence the Sun, Moon, and Earth become aligned straightly enough (in syzygy) for an eclipse to occur.
Eclipse seasons should occur 38 times within a saros period (6,585.3 days).
These distances vary because both the Earth and the Moon have elliptic orbits.
In other words, because the eclipse season (34 days long on average) is longer than the synodic month (one lunation, or the time for the Moon to return to a particular phase and about 29.5 days), the Moon will be new or full at least two, and up to three, times during the season.
Eclipse seasons occur slightly shy of six months apart (successively occurring every 173.31 days - half of an eclipse year), the time it takes the Sun to travel from one node to the next along the ecliptic.