How do messages get from the destination airport to a pilot?
When the crew got their clearance for the trip from Clearance Delivery over the radio in the flight deck, it would've included a departure time window with the delay, and they may have been told by the CD controller why, or may have figured it out for themselves because it's a routine occurrence for the airport they were headed to (like Chicago).

Or, they were told prior to that during the planning brief with Operations. All the planning and filing is done by specialists in the airline's Operations department. The crew just shows up at Operations and picks up a package with everything prepared; the route, weather, provisional loading, fuel, etc. Going to an airport with a reservation protocol in place, they would have been briefed on that by Operations. That is all provisional however, the final routing, loading, fuel etc is determined when getting ready at the gate, and also when they get their ATC clearance by calling Clearance Delivery over the VHF radio before programming the Flight Management Computer.