Is there a scriptural and/or historical reason why priests wear those white collars? When did that start and why?
When Jesus speaks to the Church at Sardis through John (in Rev. 3), he says:
Yet you have still a few names in Sardis, people who have not soiled their garments, and they will walk with me in white, for they are worthy. 5 The one who conquers will be clothed thus in white garments, and I will never blot his name out of the book of life
White is typically a sign of purity, such as when Isaiah says:
Come now, let us settle the matter," says the LORD. "Though your sins are like scarlet, they shall be as white as snow; though they are red as crimson, they shall be like wool.
Having a white collar is like having a yoke of purity around your neck. It is a symbol of the obligation that priests have to model God, and the literal stranglehold (yes, they suck!) God has around us, when it comes to modelling purity.