What do the lines "sound of vernal showers/On the twinkling grass/Rain-awaken'd flowers,/All that ever was/joyous, and clear and fresh, thy music doth surpass" mean?

In this poem, the narrator is expressing the intense joy the song of the skylark gives him. It is very typical of a Romantic poet to express both love of nature and personal emotion in verse, which is what Shelley does in "To A Skylark."
The passage about spring rains comes at the end of a series of stanzas in which the narrator compares the song of the skylark to other things: a poet, a high-born maiden, a glow worm, and a rose. In this quote, the speaker compares the bird's song to a vernal (spring) rain shower. He describes the rain falling on the grass, which makes it seem to "twinkle." The rain also wakes up the flowers. However, even as joyful, clear, and fresh as a spring shower is, it does not match the beauty of the skylark's song. The point is that the skylark's music is extraordinary because it is more beautiful and refreshing than even the spring rains.

In this Romantic poem, the speaker offers praise to one of Nature's beautiful creatures, the skylark, that sings while in flight. The particular stanza you cite could be explicated in this way:
the sound of spring showers/on rain-sprinkled grass/flowers refreshed by rain/and everything that was ever "joyous, clear and fresh", pales in comparison to the song of the skylark
The poem is an ode, and what that means is that the speaker is directly addressing the subject of the poem, the skylark. Odes are intended to praise the entity that they address. The speaker respects and admires the bird and in the stanza you have asked about, essentially tells that bird that its song is superior to other beauties found in the natural world.