The Pilot Wave Radio is your method of communication with the outside world. You can use it to receive transmissions.
There are three main sources of transmission received:
"Urgent Message!" from Rodina (the main ship, still decades from the Zorica system). Through this, the officers in charge at Rodina issue your mission profile, instructing you what to do next to save mankind. This is updated as you upgrade your ship, becoming capable of ever grander endeavours.
"Encrypted Communications" from an unknown source. This message is fully delivered to you at the start of the game, but you need to travel the Zorica system to find encryption keys to decrypt each part of the message (12 in all).
"Concealed Messages" are added in the Rodina 1.2 update and contain the main story added in the 1.2 update, because of spoiler reasons the communications are not described here.
The Pilot Wave Radio displays a blue effect when there is an unread message waiting for you.