The opening episode of Gilmore Girls makes a point of setting up the awkward and estranged relationship between Lorelai and her parents - Richard and Emily.
When Rory gets accepted to Chilton (a prep school) mid term Lorelai is faced with the large sum required as an enrollment fee along with the first term's tuition, or Rory cannot accept the open spot. Not wanting Rory to know that money for school is an issue, Lorelai discusses ideas with her friend and co-worker Sookie who suggests asking Lorelai's well-to-do parents for help. Lorelai refuses to even accept the idea (having unhappily left home at 16 to live her own life away from Richard and Emily)... until she realizes that there is no other option.
Semi unwillingly, Lorelai visits her parents (who each make a crack about Lorelai showing up on a day that isn't a holiday), explains the situation and asks for a loan. To her surprise Lorelai's parents agree. Seizing an opportunity Emily Gilmore requests that she take on an active role in Lorelai and Rory's lives by having mandatory Friday night dinners and weekly phone calls about Rory's schooling and updates about their lives until the loan is repaid. Seeing the strings attach themselves to the situation, Lorelai only asks that the loan remain a secret from Rory.
The next evening Lorelai and Rory find themselves standing on the front porch of the Gilmore home staring at the door. This is where Rory references The Little Match Girl.
In the story of The Little Match Girl we learn about a poor girl who is out in the street unsuccessfully trying to sell matches to please her unloving father. It is a cold winter and the girl is barefoot, yet she chooses to stay out in the cold rather than go back to the bad situation at home. Ultimately, the little girl freezes to death.
While the story itself isn't funny, I find that the reference to The Little Match Girl makes the scene of Lorelai standing out on her parent's porch quite entertaining. Suddenly you sense that Lorelai is like that little girl - she'd rather stay out in the cold and die than knock on her parent's door and face what's waiting for her inside. I think this was an excellent way to let viewers understand what Lorelai was supposed to be feeling!