Words in brackets refer to the subjects that were implied in the japanese lyrics (but not sung). Usually japanese language does not mention the subject because the listener knows who the subject is from context.
I think there is a punctuation error in the kanji text:
"手のひらからこぼれ落ちた君のかけらを拾い集め" -> "Pick up and gather the pieces of you that you dropped from the palm of your hands" However,
"手のひらからこぼれ落ちた。君のかけらを拾い集め" -> "[It] dropped from your hands. Pick up and gather your pieces."
The punctuation makes a difference in meaning. The first version without the period looks funny to me. I am not sure if I translated "こぼれ落ちた君" correctly because it looks strange to me. That's why I think there is an error in the kanji section.
If you find any mistakes with my english translation, please correct them. Thanks.