This is an excellent question, and one that many parents often don’t consider before signing their child up for camp.  The answers to this question are many and varied, so the most commonly cited benefits of camp are listed here.
Camps offer opportunities for fun – both in unstructured recreation and structured activities and games.
Camp provides an environment for children that frees them from technological distractions and asserts the importance of interpersonal relationships and cooperation.
Camps are excellent places for children and youth to improve athletic, artistic, and intellectual skills in a supportive and non-competitive setting.
The community-oriented nature of residential camp allows campers to more easily make friends and develop social skills.
Camp can further self-esteem and self-confidence by providing experiences in which campers challenge themselves to the extent they feel comfortable – as individuals and as part of a group.
Group-building initiatives and games are instructive in illustrating that failure is useful as a learning experience and can be turned into success through critical thinking and trust within a group.
Small-group camping also emphasizes the development of skills and traits gained from having ever-present, positive adult role models.
For an excellent treatment of the psychological benefits of summer camp, please read Homesick and Happy: How Time Away From Parents Can Help a Child Grow, by Michael Thompson, Ph.D.