I've begun reading the Scientific American special issue on Time.
"From the fixed past to the tahgible present to the undecided future, it feels as though time flows inexorably on. But that is an illusion." So writes Paul Davies in That Mysterious Flow.
Nothing in known physics corresponds to the passage of time. Indeed physicists insist that time doesn't flow at all; it merely is.
Einstein famously wrote... "The past, present and future are only illusions, even if stubbon ones." Einstein's startling conclusion stems directly from his special theory of relativity, which denies any absolute, universal significance to the present moment. According to the theory, simultaneity is relative.
As to why our brains think time exists, Davies suggests that maybe it's due to the irreversible process of entropy. Things get messier over time; they never un-mess. New memories add to the brain's entropy. Maybe we experience this as the passage of time?
