I often hear about "Vasudhaiv Kutumbakam" which means "The whole world is one family".
What is the story behind this quote?
There is a detailed discussion in the Indian-American scholar Rajiv Malhotra's book, Indra's Net'. I am posting here an excerpt from Rajiv Malhotra's book:
The most common Sanskrit phrase quoted in this regard, often repeated at popular gatherings, is vasudhaiva kutumbakam', which means, 'the world is one family'. This assertion is recited as a signature of Hindu benevolence towards others and is often used to promote a spirit of unconditional generosity towards others. The earliest occurences of 'vasudhaiva kutumbakam' are found in the Hitopadesa and the Panchatantra, which are collections of fables discussing practical situations in life through talking animals. These stories are meant as an entertaining educational aid for children, and their popularity has carried them to distant places across Asia and Europe. It is important to note that there fables impart values that are pragmatic and contextual, as opposed to high philosophy.
In these stories the phrase is used in both ways - to advocate harmony towards others and also to advocate vigilance and suspicion towards those one does not fully understand. The moral of the story depends on the context. If one is operating from a position of power over others, and those others have good intentions, then the message is to include them with mutual respect. On the other hand, if one is ignorant of others' intentions, or one is operating out of weakness, then such behaviour is seen as a mark of foolishness. It is by no means a blanket statement of an unconditional welcome as is often made out in popular usage.
In one story in the Hitopadesh, a cunning jackal, trying to create a place for himself in the home of a native deer says, 'vasudhaiva kutumbakam' in his appeal to the deer. The deer ignores warnings from other animals, who caution that it is unwise to trust someone at face value without first ascertaining his history, nature, and intent, Upon deceitfully acquiring the deer's trust and moving in his home, the opportunistic jackal later tries to get the deer killed. Indeed the moral of the story is that one must watch out for cunning subversives. Blindly trusting those who preach 'universal brotherhood' can lead to self-destruction.
Hindus follow vasudhaiva kuṭumbakam (whole world is indeed one family) because that's broad minded thinking. It is a proverb that goes like below:
ayaṃ nijaḥ paro veti gaṇanā laghucetasām
udāracaritānāṃ tu vasudhaiva kuṭumbakam [Panchatantra - 5.3.38]Meaning
Considering "This person is ours (friend) and this person is not" is narrow minded thinking. But for the liberal broad minded person the whole world is indeed one family.
It is a proverb so we can't say what's its origin or story. But being a proverb it has been used in story books like Panchatantra and Hitopadesha. The Mahat Upanishad(6.71) also mentions it in a slightly different form.
REFERENCE: Vasudhaiva Kutumbakam
I don't know if there is any story behind quote or not. But, according to my understanding, it has a philosophical meaning. According to the Advaitha Vedanta, all the jivatma (all the lives including plants, animals and humans) were a part of the Paramatma. Because of Maya, they have been born in this world in any one of the life form. According to the Karma that the jivatma gathers in its life, it is either reborn or attains Mukthi (union with Paramatma).
Having said that, all the life forms (not just the human beings) trace back their origin to the Paramatma. Paramatma is the source and destination of all jivatma. Hence, all the jivatma on earth are considered to be from the same family (that of the Paramatma).