As we first see Snow White, she's cleaning the steps of the castle. She moves over to the Wishing Well and begins to wish (in song, of course) that "the one I love will find me today." As the Prince overhears this, he climbs over the wall and into her property. Moments after this, she runs inside, though the Prince continues singing a love song to her. In less than three minutes, she's head-over-heels. To make matters worse, to the best of my knowledge, Snow White is merely fourteen in this version (correct me if I'm wrong).
Snow White knows nothing about the boy, except that he is, indeed, a prince. Once she finds shelter with the dwarfs, she tells them a story, beginning with the quote above. Snow White goes on to say that someday her prince will come, "and away to his castle, we'll go. To be happy forever, I know." Before being poisoned by the apple, her wish is that they will live happily ever after.
Once Snow White is poisoned, she's still beautiful after death. The dwarfs place her in a glass coffin to still admire her beauty. The Prince hears of this, and comes to see her, and he sings his wonderful love song again.
After his song, he kisses Snow White's lips and she wakes up to see the Prince there for her. Without a word, he carries her away to his horse. After giving the dwarfs a kiss and a goodbye, they're off, riding into the sunset.
So, overall, The Prince eavesdrops her singing, enters her property uninvited, and kisses a corpse; yet, he manages to steal her heart. And, better yet, she thinks of him as "romantic."