Probably when I heard about (as it was then) this new fangled "internet" thing going around.

Suddenly computers weren't isolated individuals, or networks, but had the power to talk to *other* computers as well, and exchange information with each other, and then take decisions and action based on that information - even talking to other computers in turn!

The potential for what this could do blew my mind a bit as a kid - and I'm not sure anyone, myself included, even then understood anywhere close to what the full impact would be in a few decades time.