What is the meaning of Hare Krishna mantra, how does chanting Hare Krishna mantra helps anyone?

The first principle in devotional service is to chant the Hare Krishna maha-mantra (maha means “great”; mantra means “sound that liberates the mind from ignorance”).

Krishna or Christ- the name is the same. The main point is to follow the injunctions of the Vedic scriptures that recommend chanting the name of God in this age. The easiest way is to chant the maha-mantra: Hare Krishna, Hare Krishna, Krishna Krishna, Hare Hare/ Hare Rama, Hare Rama, Rama Rama, Hare Hare.

Rama [O Supreme Enjoyer] and Krishna[O all-attractive Lord] are names of God, and Hare [is the energy of God]. So when we chant the maha-mantra, we address God together with His energy. Chanting is a prayer to Krishna, so at the beginning of the Hare Krishna maha-mantra, we first address the internal energy of Krishna, [Hare] that means “O Radharani! O, Hare! O energy of the Lord!” When we address someone in this way, he usually says, “Yes, what do you want?” The answer is, “Please engage me in Your service”.

Just as there are males and females in the material world, similarly, God is the original male (purusa), and His energy (prakrti) is the original female. This energy is of two kinds, the spiritual and the material. At present we are in the clutches of the material energy. Therefore we pray to Krishna that He may kindly deliver us from the service of the material energy and accept us into the service of the spiritual energy.

In the Narada-pancaratra it is stated that all the Vedic rituals, mantras, and understanding are compressed into the eight words Hare Krishna, Hare Krishna, Krishna Krishna, Hare Hare. Similarly, in the Kali-santarana Upanishad, it is stated that these sixteen words, Hare Krishna, Hare Krishna, Krishna Krishna, Hare Hare/ Hare Rama, Hare Rama, Rama Rama, Hare Hare, are specially meant for counteracting the degrading and contaminating influence of this materialistic age of Kali.

This chanting is exactly like the genuine cry of a child for its mother’s presence. Mother Hara helps the devotee achieve the Lord Father’s grace, and the Lord reveals Himself to the devotee who chants this mantra sincerely.

God has kindly made it very easy for us to chant His names, and He has also invested all His powers in them. Therefore the names of God and God Himself are identical. This means that when we chant the holy names we are directly associating with God and being purified. Therefore, we should always try to chant with devotion and reverence. The more attentively and sincerely you chant these names of God, the more spiritual progress you will make.

Source: A.C. Bhaktivedanta Swami Prabhupada (2014 edition), “The Science of Self-Realization”, Page 17, 109, 129 and 170.
A.C. Bhaktivedanta Swami Prabhupada (2012 edition), “Message of Godhead”, Last Page.