Why was the Battle of Lake Regillus fought?
First of all, please realize that it is not completely clear as to whether the Battle of Lake Regilius ever actually happened. Records from the time are poor and there are clearly mythical elements to the stories that have come down regarding this battle.
If the battle did occur, it happened because of a power struggle in the area of Rome. This was in the 490s BC, before Rome came to dominate all of the Italian Peninsula. There was a confederation of cities in the area of Rome that is now known as the Latin League. The Latin League and the Romans periodically fought for power. This battle happened (if it really did) as part of a struggle for power between Rome and the Latin League.