One day Lord Ganesha was returning home after enjoying a fabulous meal fed to him at Lord Kubera's house. Kubera the lord of wealth was a big devotee of Lord Ganesha had prepared all of Ganesha's favorite foods including varieties of Modak, laddus, chakkuli, kadubu and much much more.
Ganesha's stomach was soo full that he was fiinding it difficult to even walk. So he calls his vehicle "the mouse" to carry him back home. When the king mouse sees how big Ganesha's stomach has become he starts thinking "oh! lord, my master has done it again.. it is such a long distance back to home, how will I carry him on my back".
Ganesha sat on the mouse and the mouse started dragging himself back home. On the way when they were passing through a forest, Gaensha's stomach burst open and the laddus and modakas started falling out and Ganesha fell off the mouse on to the ground. Ganesha quickly saw around to see whether anyone saw his fall . Suddenly he hears a loud laughter.. The very embarrassed Ganesha suddenly realises that the Moon in the sky has seen him and is laughing at his plight.
The Lord searched around to for something to tie around his stomach before getting up from the ground. He saw a snake crawling by and uses it as a belt to tie his stomach. This is why even today we see the snake around Lord Ganesha's stomach. Holding it in.
The angry lord, looks up at the moon and curses him to oblivion. The Moon is now scared and comes down to earth and falls on Lord Ganesha's feet and begs for forgiveness and tells him he will undergo the waxing and waning cycle once in 15 days to repent for his foolishness. Lord Ganesha forgives the Moon and tells him "O Moon, you laughed at my discomfort so whoever sees you today will have to face accusations and blame".
So once the bright and shiny moon becomes dark and lustreless and seen only twice in its full form every month. and this day, the 4th day in the month of Badrupada Shukla paksha is celebrated as "Ganesha Chaturthi".
Note: There are different versions of this story and I am telling you the one that I have heard. Enjoy :)