Whereas the Twos act as reconcilers, the Threes are more mediators, helping to communicate and build relationship. As mediators, the Threes are the one between two, or from the two. Seen through a lens of the Golden Dawn's use of the Tree of Life, the Threes are still in the supernal realm above the Abyss, but about to cross the Abyss to the more manifest realms. They can be viewed as holding all the divine potential of an ideal preparing to manifest. The threes are about to set off on a journey, and can help mediate between the unmanifest and manifest. The Threes are associated with the Sefirot Binah, "Understanding".
The Three of Wands is the creative potential, Cups the emotional potential for joy, and Disks the potentials of one's true work in the world, one's vocation. The Swords follow suit, but in an unusual way. Beyond the Golden Dawn astrological association of Saturn in Libra (a tough combo), it is the knowledge of losing knowledge, the intentional ignorance we place upon ourselves by coming into this world in order to grow, learn, and beget novelty. It is the loss of the higher divine perspective for the potential to become human.
Sun in Aries
The Three of Wands relates to one's sense of self-potential--one's skills, temperament, desires. This is what one brings to the table, so to speak.
Mercury in Cancer
The Three of Cups is the toast that starts the celebration, the building and gathering of joy that may fuel the coming trials.
Saturn in Libra
The Three of Swords is a sense of loss of one's higher knowledge, a necessary prerequisite for becoming embodied. It may also indicate other instances of soul loss that may need to be reclaimed.
Mars in Capricorn
The Three of Disks is the skill one brings to a situation, one's work in the world apart from any particular job or employment one might have.