What you have to do is earn enough money to level up your house at least two times, have a person up to at least 8 hearts, and some say that you need a double bed, which you can get after you get one heart with Kurt, Woody, or Joe. On the next sunny day the mayor will come to your house and say that there's a rumor you've been doing well around the village. Then he'll go on about getting married, and eventually come up with the subject of the blue feather. After you're done talking to him, a cutscene will show you at the mountain (I forget the name) and he'll tell you to climb the mountain. If you succeed, another cutscene will show you finding a blue feather. And after all that stuff, it will still be 6 AM, you'll be down the mountain talking to the mayor again, and confetti and such, and then you can propose. Whether he/she accepts depends on their heart level.
By the way, this can happen any sunny day, regardless if it's a festival day or not.
Omg, that was long.