The European skylark sings only when in flight. When the speaker of the poem starts praising the bird, it is already out of his sight. So, the speaker only hears the bird, giving it a spiritual (unseen) quality. The bird is compared to a spirit or a soul which has left the physical bounds of earth, but whose song, or spirit, can still be heard:
Like a star of Heaven
In the broad day-light
Thou art unseen,--but yet I hear they shrill delight,
A star can not be seen in the "broad day-light" but we know it is still there. Here, Shelley conflates a natural phenomena (a star invisible in the daylight or the heard but unseen skylark) with the idea of a heavenly presence.
As the skylark flies "Higher and higher still," this is symbolic of a being escaping the physical constraints of earth, essentially becoming like immaterial: a spirit.
The speaker claims that the skylark's song is more "bright" (in the sense of being vivid and alive) than the drops from "rainbow clouds." Comparing visual and audible imagery, the skylark's song is more affecting than visible beauty.
There is more symbolism utilizing this notion that the effect of one sense is absent but another sense dominates the speaker's perception. For example, the rose is "embowered / In its own green leaves--" meaning it is enclosed and therefore invisible. But its sweet scent is carried on "warm winds."
The skylark is described as "blithe" in the opening line. This means happy and without care or concern. The speaker longs to be free of life's concerns and fears; to be like the skylark, not worried about fear, mortality, etc.
Yet if we could scorn
Hate and pride and fear;
If we were things born
Not to shed a tear,
I know not how thy joy we ever should come near.
The skylark, more particularly its song, symbolizes ultimate joy. The speaker, Shelley in this case, asks the skylark to teach him this joy in order that he might give his poetry the same blithe quality of the skylark's song.
Teach me half the gladness
That thy brain must know,
Such harmonious madness
From my lips would flow
In the end, the song is comparable to poetry itself and is therefore symbolic of the potential of poetry to be as effective as the unseen but heard song. There is the posthumous indication that like the unseen but heard song, Shelley's own poetry will be heard (or read) after he has died. Therefore, the skylark's song is symbolic of the beauty in nature as well as the beauty in poetry.