When you enable airplane mode you disable your phone's ability to connect to cellular or WiFi networks or to Bluetooth. This means you can't make or receive calls, send texts, or browse the internet.
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In situations where a power source isn't available, turn airplane mode on to stretch your device's precious battery life. Just remember that airplane mode disables your access to cellular data, so don't expect to be able to send text messages or make cell phone calls in this battery-saving mode.
If he put it on flight mode, put it in his pocket and never took it out while you were together, it's likely he just doesn't want interruptions to his time with you. Even then, it would be more common to put it on silent, so that he could check from time to time that he could see who had been contacting them.