Ganesh Chaturthi (Chauth) is the day for worshipping Ganpati Maharaj. In the Holy Shikshapatri, Bhagwan Swaminarayan urges His Acharyas to worship Ganpati Maharaj on this day (Bhadarva Sud Chauth).
The leela behind the vrat is that on this day, Ganeshji was on his way to Kailash and Chandradev (Moon God) started making fun of Ganeshji. At first Ganeshji tried to ignore Chandradev, but eventually he became very angry and cursed Chandradev that no one would look at the beautiful moon on this day and that those that do would be cursed with misfortune.
On hearing this Chandradev became ashamed of his actions and went into hiding. The devtas were concerned by this and went to Bhagwan to ask for help. Bhagwan advised that the curse was given by Ganeshji so only he could retract it. However, Bhagwan showed the devtas, including Chandradev and of pleasing Ganeshji and thereby convincing him to retract the curse.
The devtas made a golden statue of Ganapati Maharaj and worshipped it for the first four days of Bhadarva Month with flowers, sandalwood, etc. They presented thaal of ladoos and performd ektaanu (ate only once a day). Then on the fourth day, Ganeshji was pleased and retracted his curse and Chandradev was able to show his face again.
In addition to lifting the curse Ganeshji declared that those who worshipped him on this day in this way would find happiness and good fortune.
Hence people now worship Ganeshji on this day.