It takes 12 years for Guru (Bruhaspati, Guru of Devas) aka Jupiter to complete its 1 year (one revolution around sun).
According to Srimad Bhagvatam 5.22.15:
Situated 1,600,000 miles above Mars, or 10,400,000 miles above earth, is the planet Jupiter, which travels through one sign of the zodiac within the period of a Parivatsara. If its movement is not curved, the planet Jupiter is very favorable to the brāhmaṇas of the universe.
Here Parivatsara is one of the name of a year. It's stated Jupiter travels one zodiac sign in a year. And as we know, there are toal 12 zodiac signs. So, it takes 12 years for a Jupiter to complete 1 revolution around sun.
Just as humans celebrate birthday every 1 human-year, so we celebrate divine festivals like kumbh-mela every 1 Bruhaspati-year.
Also, see 12's significance in comment posted by @Ajay above which links to yahoo answers.
From Mahabharata Vana Parva chapter 134, Ashtavakra says importance of each number. In regards to twelve, he says
Ashtavakra said, Twelve months compose the year; twelve letters go to the composition of a foot of the metre called Jagati; twelve are the minor sacrifices; and twelve, according to the learned, is the number of the Adityas.'
Above link has more details on importance of number twelve.