Passing progressions in football are the patterns that a quarterback must read through when making a decision as to who they should throw the ball to. They must decide who to throw the ball to as well as how fast to do so.
A quarterback before the play begins knows where they want to throw the ball. That can be based off of a matchup advantage or the route that they are running. They select a primary receiver that they want to throw the ball to. If they aren't open then they progress down their list to the second receiver (secondary option). That continues throughout all receivers if no one is open. The quarterback also has safety of checkdown options to prevent taking a sack in the event no receiver gets open.
Not only is it picking the player to throw to, but the quarterback must also decide if it's a play action, quick throw, long throw, or anything in between.
If no receiver gets open it is important to have a backup route. That's where the safety valve comes in, they do not run a route far from the quarterback. They give the quarterback an option a couple yards down field.
An audible is when the quarterback changes the play at the line of scrimmage, after the huddle is broken. They might change who the primary receiver is or change the blocking scheme. A large part of passing progressions is having that safety option; a quarterback can give themselves another one with an audible.
One thing quarterbacks audible at the line of scrimmage is a hot route. If the quarterback believes the blitz is coming they might have the running back run a short route to be able to dump the ball off. It is very similar to how a wide receiver can be a safety valve.
When a team has a plan and knows where they want to throw the ball, their offense becomes a lot more efficient.