The pilgrimage begins in the spring, "whan that Aprill, with his shoures soote / The droghte of March hath perced to the roote" (General Prologue 1 – 2). Since this is the beginning of the poem, and the beginning of the pilgrimage (which itself is the beginning of repentance), it's likely that springtime here is a symbol of beginnings. And the beginning of things is exactly what the poem emphasizes in its description of springtime, talking about how the wind spreads the seeds that peek their heads above the soil as they begin to grow into crops, and how birds begin their mating season.
This brings us to another thing that springtime symbolizes: sexuality. You see it in the way April is piercing March "to the roote" with his showers, watering things and causing them to grow in the same way a penis "waters" the ovum and causes it to grow. In its masterful opening, the poem links springtime and sex in the way that they both cause new life to begin.
The poem might start this way in order to remind us how pilgrimages are also a start of new beginnings. See, the idea of a pilgrimage is that you start on a journey of repentance, beginning a new life, one free from sin. In the beginning of the poem, then, the springtime is a symbol of the new beginnings and the creation of new lives the pilgrims are about to undertake.