There are many explanations of the Mahamantra by our different acharyas, Shat Goswamis, Jiva Goswami, Rupa Sanatan Goswami and Bhaktivinod Thakur, so many.
But we are very well versed with the explanation of Srila Prabhupada’s explanation to Hare Krishna Mahamantra, He said that ‘Hare’ is the address to Mother Hara and Mother Hara is Srimati Radharani, pleasure potency of Krishna.
So ‘Hara’ becomes ‘Hare’ when we address, this is Sanskrit grammar, Sita becomes Site , Radha becomes Radhey.

So Hare is the address to Srimati Radharani, Mother Hara. And we are praying to Mother Hara, “My dear Radharani, please engage me in Krishna’s transcendental loving devotional service.”
That is the meaning of Hare Krishna Mahamantra. That please Radharani give me transcendental loving service to Krishna.

We are supposed to serve Krishna, throughout the day, 24 hours a day , eternally. So please I have forgotten my constitutional position to serve Krishna, please engage me Krishna’s transcendental loving devotional service.
And why do we want to get engage in His service? Because if we serve dog, dog will love us, if we serve God, God will love us.
We want to serve Lord Krishna, so that He can love us, which is the topmost goal.
“He can love us , we can love Him”.
If we love somebody, first thing we do is service. Without service, there is no question of love.
This is the meaning of Hare Krishna Mahamantra. That is why we propagate this mantra so that people can chant, get rid off all the sins, become pious and ultimately they develop the pure love for Krishna. When we develop pure love for God, Krishna, then automatically we love His creation, such as birds, beasts, cows, human beings, trees, Demigods.