Usually someone who was 'raised' by the internet. Or someone who has been on the internet for a very long time.
In some cases Big On Internet people's days revolve around the internet, and devices related to it.
Big On Internet is also referred to as BOI.
Person 1: "Why does Axle keep making references to things on the internet?"
Person 2: "He's Big On Internet. He's been on the internet since nightcore. He basically has a long-time relationship with the internet."