The word “pilgrim,” derived from the Latin peregrinum, conveys the idea of wandering over a distance, but it is not just aimless wandering. It is a journey with a purpose, and that purpose is to honor God.
After the death and resurrection of the Incarnate God and the spread of Christianity, adherents felt a longing to tread in the footsteps of their Savior, His Holy Mother, and His chosen followers, the holy Apostles. Even in the early centuries, when millions of Christians were martyred for their Faith, the faithful flocked to the tombs of favorite saints to venerate their remains, sometimes at the risk of being martyred themselves. What were their motives? Knowing that in honoring His saints, they were honoring God Himself. Some pilgrimages were done in penance for sin; some were done in petition for a special blessing or favor; and some were undertaken simply out of devotion.
One of the earliest usages of the word is found in the writings of Saint Augustine of Hippo (354-430). His work, Peregrinatio, described a Christian spiritual journey as a kind of self-imposed exile of the pilgrim in which he searched for God’s Truth in his wanderings while visiting the holy shrines of the Faith.
Excepts taken from Catholicism.org article found here.
Here is another great article about the purpose of a pilgrimage from Our Sunday Visitor.