The idea of sacred travel runs deep in religion, dating back to when early humans would climb hilltops to be closer to God, or got to a specific spot to dance around in circles. People go on pilgrimages to bring themselves mentally closer to God, and to strengthen their faith with God. Many people believe that by going on a pilgrimage they will do penance (making up for sins). Some just want a religious experience, go on one for thanksgiving, or want to show devotion by making an effort and possibly suffering.
To strenstrengthen their faith with god
Because it helps them in their Christian journey. It may make their faith be strengthened to go to holy places.
Because they want to see where jesus walked and performed his miracles...
Because they can empathise with God and Jesus, experience there suffering ect. It might cure them physically or help them to deal with their illness.