http://www.songfacts.com/detail.php?id=762
Apparently the people from this website read articles from Chris Martin (lead singer) himself. So, this is what they had to say.
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Lead singer Chris Martin has said that the lyrics are about being devoted to someone, wanting to do anything for them - writing a song for them, swimming across the sea for them and so on. He said that it's not necessarily a romantic devotion, but someone that you look up to and would do anything for, a brotherly love. (thanks, Adam - Sussex, England)
The band came up with the line "Look at the stars, look how they shine for you" after looking up at the night sky. They were recording in Wales, which had a nice, clear sky.
The color yellow can have many negative implications ("Yellow Journalism," "Yellow Fever," a "Yellow Card" for a foul in soccer), but for Coldplay, it is a beautiful color. According to lead singer Chris Martin: "It was simply because that word sounded nice, it just seemed to fit, no other reason. None of the other colours would have sounded right really!" The Beatles were kind to the color when they used it in their song "Yellow Submarine."
Martin initially wrote the words and melody, then the rest of the band wrote the other bits and pieces (the hook, the riff, fills etc.). (thanks, Edward Pearce - Ashford, Kent, England)
"Yellow" is a term that can mean cowardly or afraid. From this standpoint, the singer could be afraid and nervous to show his love, until he "ran across" and "jumped across" for her. He finally got the nerve to express his love. (thanks, Stefan - Winona, MN)