Memory accretion explains sense of flow
The process of memory accretion gives rise to our sense that time flows. This explanation presumes that we can remember not just specific events and their temporal ordering but also the process of memory accretion itself - see citation from Julian Barbour for a similar (timeless) account.
We may remember precisely when we got up this morning because we were woken by an alarm at a specific time. We may remember roughly when we ate breakfast, and so on. Recent research has shown that the brain has a number of different time-keeping mechanisms specialized to different durations (from the circadian rhythm to the 'moment of experience').
We also presumably remember the process of memory accretion itself - that the stock of time stamped events that can be accessed in memory is being constantly added to. From this we infer inductively that the process will continue - that there will be future events, not yet realized. The accretion of memories also accounts for why past events cause less concern or anticipation than future ones - the fact they are encoded in memory shows that they are realized, finished - 'done and dusted'.
Physicist Julian Barbour in his book The End of Time cited below makes a similar point. The main thesis of the book is that time itself is superfluous - and he claims to show how eliminating from physical theories makes possible a reconciliation of quantum mechanics and general relativity. But how does how account for the subjective experience of time? He puts forward the notion of 'time capsules'. At all times, we have encoded in our brains information analogous to multiple stills in a movie - information that the brain uses to create an impression of movement or change over time.