In the first stanza of the first section, the speaker refers to the "breath of Autumn's being" and although the wind is described as "unseen," the speaker can see the effect of the wind's presence with the dispersal of dead leaves. The leaves are driven to their "wintry bed" where they become part of the soil and part of the regeneration of life in the spring.
The personified Spring will sound her clarion (trumpet) in the Spring, signaling spring's renewal of life. This use of clarion is a metaphor but one could interpret this as literally representing the sounds of spring: birds chirping, green leaves on the trees rustling in the wind, and the rain (April showers) which comes from the "vapours" (clouds) formed during this time.
In the third section, the speaker notes another sign of spring's renewal: the scent of flowers:
All overgrown with azure moss and flowers
So sweet, the sense faints picturing them! Thou
For whose path the Atlantic's level powers
The flowers are "sweet," and "sense" can refer to the smell and the image (smell and sound) with the pun on "sense" and "scents."
In the final stanza, the speaker asks the West Wind to inspire him, to "move" him or to move his thoughts (poetry) so that others may hear/read it. This is comparable to Wind moving the leaves to all areas of the landscape where they will go into the soil and be reborn in the spring. In this stanza, the speaker wants to feel (be touched) by the power of the wind, but this is metaphorical as well. He wants to feel the inspirational power, thus he wants to be emotionally and creatively moved by this notion of regeneration. He also hopes that his poetry will be reborn (as buds are in the spring) and read by future generations.