The Airstrike mode is similar to the Airstrike Command Card that was previously available. When you select an Airstrike you set your target and the angle of the airstrike, which you would use to align with the direction of the ship you are targeting to get maximum effectiveness. The difference is that now enemy ships can defend against airstrikes, so depending on the number of flights they have set on Air Defense and the number of AA guns they have installed, your attacking flight could be shot down and/or damaged.
The Air Defense mode will give your flight orders to intercept any enemy airstrikes within a radius around your carrier. This doesn't guarantee that it will stop an enemy airstrike but it will at least damage their attacking flight.
You can also arm your ships with AA guns (during the Turret selection menu) which will automatically fire on attacking aircraft. A combination of AA guns and Air Defense flights will go far to keeping your fleet safe from enemy airstrikes.