I like the famous song. But I can't understand what does "to promise the earth" mean. Does she want to give the Earth to him? I suspect native English speakers find it a strange question. Maybe I don't know all meanings of 'promise'. Could anyone explain to me this line?
This is a standard idiom in English. The earth here is a metaphor that symbolizes the impossible, the immoderate, or the unfulfillable:
promise (someone) the earth (or moon)NOAD
make extravagant promises to someone that are unlikely to be fulfilled:
interactive technology titillates, promises the earth, but delivers nothing
The origin of this idiom is probably from the third temptation of Jesus (in the Bible), where the devil tempts him with the ridiculous promise of all the kingdoms of the world:
Again, the devil taketh him up into an exceeding high mountain, and sheweth him all the kingdoms of the world, and the glory of them; And saith unto him, All these things will I give thee, if thou wilt fall down and worship me.
— Matthew 4:8–9 (The Bible, King James Version)