There's at least 3 sources: Mistletoe was a symbol of fertility to the Celts in England (pre-Roman), because the leaves remained green in the winter, and so was seen to have life-giving powers. In Medieval times it was hung over a doorway to ward off evil spirits, and in the late Renaissance in England it was fashioned as a "kissing bough" and placed in the main room of a house during Christmas.
The ancient Greeks saw mistletoe as a symbol of peace, and it was common practice at a wedding for couples to kiss under the plant.
Finally, in Viking lore, Balder, god of light, had a dream about dying. His mother, Frigga, goddess of love and fertility, asked all the elements of the earth to spare him. They agreed because everybody loved Balder, and in celebration all the gods jovially tried and failed to kill him with things made from the earth (those Vikings!)
Loki, god of mischief, found out that mistletoe was exempted from the list because it was so insignificant (ie it grows on a tree, not in the earth). So he played a "prank" and shot Balder with an arrow made of poison from a mistletoe berry, and he died.
Frigga struggled and finally saved her son (in a happier version of the legend), and in joy she give mistletoe a more noble place in nature and announced that she would bestow a kiss for good luck and protection for anybody who passed under it.