I don't agree with that statement though. Extra attacks do not say anything about having to be on your turn. They just say whenever you make an attack action you may do another attack. If you are readying an attack action , you should get every attack. Otherwise there is no incentive for fighters to ever Ready an attack.
Sure it does.
EXTRA ATTACK
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
I mean I missed it too, but there it is in the rules clear as day. A character with extra attack might be less inclined to ready an action but it is still sometime the only way to deal with some creatures, like flyers if you don't have a ranged weapon.