Reason One: because his Lily-pad didn't get enough love from her sister

James Potter grew up an only child. After eleven, Lily Evans seemed to too. James had heard Remus talking to her one day, after she had left the Great Hall crying and didn't show up to lessons. Her sister (some annoying cow named Petunia) had apparently called her a freak and didn't want Lily to come home for Christmas that year. They had been in their third year, and at the tender age of thirteen he was smitten. He already knew that his dearest Lily was a rare and beautiful flower- one that he would marry- and because her sister didn't like her he was all the more determined to show her that he did. Sometimes he honestly wished he could meet this Petunia, just so he could give her a piece of his mind. However he refrained, because he knew it would make Lily hate him even more. So he decided if he couldn't make Lily's sister show her love, then he'd have to make up for it in another way. So James decided to ask Lily out.

"Hello Lily! You look particularly smashing today! Would you care to accompany me to Hogsmeade this weekend?"

"No Potter! Why on earth would I want to go with a bullying toerag like you?"

So maybe his first attempt didn't go quite as well as he had hoped, but he didn't worry. James knew he had four more years to show her exactly how much he could make up for her sister.