What objects are the fountains Of thy happy strain?
The speaker keeps prying into the bird's secrets. He wants to know what the source, the origins ("the fountains") of his happy melody ("strain") might be.
This is all about trying to understand the meaning behind the beauty of nature, to use comparisons to try to get at the truth of the perfect art the speaker sees all around him.
What fields, or waves, or mountains? What shapes of sky or plain?
What inspires these melodies? Is it other things in the natural world (on earth, at sea, or in the air)? Is the skylark singing about the world it sees?
What love of thine own kind? what ignorance of pain?
Or maybe it's the feelings of the skylark that make it sing the way it does. Maybe it sings because it loves another skylark (its "own kind").
Maybe it sings with the joy of never having known what pain feels like.
That last phrase is especially great. We can feel all the sadness and longing of the speaker in that "ignorance of pain." He is dreaming of a world beyond pain, a creature that can make art without any of the suffering that humans feel. It's kind of heartbreaking. (But then again, we're kind sappy.)