Swati means independent, or self going. The symbol is a young plant bending in the wind: flexible enough to survive a storm, not yet rooted deeply enough to ignore one.
In Hindu mythology, Vayu is the god of wind and of prana, the breath that keeps a body alive. He is also the father of Hanuman, which is why Hanuman worship is one of the standard remedies for this star. Vayu moves, circulates and refuses to be contained, and every one of those qualities shows up in Swati natives.
There is a well loved belief attached to this nakshatra. A raindrop that falls while the Moon is in Swati is said to become a pearl, if it lands in an oyster. The same drop becomes camphor on a certain plant, and poison in a serpent's mouth. The same drop, three different outcomes, depending entirely on where it lands. That is a good description of what this star does.
Its shakti (the special power a nakshatra carries) is scattering. Wind disperses things. That can free something that was stuck, and it can also leave it without direction.
Swati is known for independence, trade and negotiation. Natives dislike being controlled far more than they dislike being disagreed with, and many of them end up self employed for exactly that reason. Saraswati, whose name echoes this star's, is also commonly associated with it in devotional practice.