The wisdom is that by shaving off one’s hair, one demonstrates one’s sincerity and humility to Allah without caring for one’s physical appearance.
Elaborating more on this, we cite what Sheikh Sayyed Sabiq states in his well-known book, Fiqh As-Sunnah:
This practice of shaving or clipping the hair is affirmed by the Qur’an, the Sunnah of the Prophet, and the consensus of the Muslim scholars.
Almighty Allah says, (Truly did Allah fulfil the vision for His Messenger: ye shall enter the Sacred Mosque, if Allah wills, with minds secure, heads shaved, hair cut short, and without fear.) (Al-Fath 48:27)
Both Al-Bukhari and Muslim quote the Prophet (peace and blessings be upon him) as saying, “May Allah bless those who shaved (their head during Hajj). “The Companions asked, “O Messenger of Allah, what about those who cut their hair short?” Thereupon, the Prophet (peace and blessings be upon him) said, “May Allah bless those who shaved. “They repeated their question the third time, and again the Prophet said, “May Allah bless those who shaved. “When they asked him for the fourth time, “O Messenger of Allah, what about those who clip their hair short?” He said, “And (may Allah bless) those who clip their hair short.”
Shaving here means removing hair of the head with a razor, etc., or plucking them out. It suffices one, however, to remove only as many as three hairs. And cutting the hair short means cutting the length of one fingertip of the head’s hair.
However, the issue of which one is obligatory, between shaving or cutting hair short, is controversial among jurists. Most scholars hold that shaving or cutting the hair short is obligatory, and one failing to do so is to slaughter an animal in order to atone for this omission. According to the Shafi`i school, it is an integral part of Hajj.
As for the second question about the reason why people go to Hajj more than one time, We should know that Hajj is only obligatory once in one’s lifetime, if he or she is physically able and can afford the journey. You can for Hajj after performing it for the first time, but this will be counted as a supererogatory act and not as an obligatory one.