Bamboo is seen as the ideal gentleman, who bends before the winds of change, but never abandons his ideals; just as bamboo is bent low by strong winds, but does not break. Like bamboo, the gentleman regains his upright position as soon as the storm has passed.
Pines are revered for their great age and strength. They are also valued because, like bamboo, pines remain green during the winter.
The plum in this trinity is Prunus mume. This plum is known in the English speaking world as the "Japanese Flowering Apricot". In fact, it is neither a true apricot nor a plum, but, like apricots, plums, cherries, peaches, nectarines, and English Laurel, it is a member of the prunus family. Prunus mume is valued both because it is early blooming and because it is the longest lived of the flowering fruit trees.
The Chinese have traditionally valued age very highly and a gnarled plum tree next to an ancient pine symbolizes friendship for life. While the individual culms of bamboo don't live very long, the Chinese have known for many centuries that one bamboo plant can form an entire grove. Bamboo plants die after flowering, but many of them only flower once in a century, so they are also valued for their age.
The ideal Chinese man was to be as pure as a plum blossom, as strong as a pine, and as pliant as bamboo.
The Three Friends of Winter are also connected with the Buddha, Confucius, and Lao Tzu.
Lao Tzu was believed to have been born under a plum tree, so it is natural that he would be represented by one.
The Buddha died in a grove. This probably didn't happen in a grove of pine trees, but pine groves reminded the Chinese of this event, so the Buddha was represented by a pine tree.
Confucius was represented by bamboo because the Chinese saw the underground root system of a stand of bamboo, in which the apparently individual stems are in fact all part of one plant, as an example of a peaceful cooperative society.
In traditional Chinese gardens, plants are chosen because of their symbolic meanings and their ability to evoke passages in literature, rather than their horticultural rarity, but I suspect that bamboo, pines, and plums were first grown in gardens for their beauty, and that the symbolism arose from their contributions to the winter beauty of gardens.
