I mean, we've all talked about the Mewtwo quote so I'll skip that.
I actually had to think about this one because specific dialogue rarely sticks with me the same way overarching stories do. But I think
"No matter where you go, everyone is connected" from Serial Experiments Lain is a good one. It is simultaneously a statement that you're never truly alone (for better and for worse) and a reminder that no matter how isolated you think you are, or how above it all, your actions will have far reaching consequences for other people. It's really encapsulates the modern world... which is very impressive since the anime is from the 90s.
"We have to change our way of thinking if we want to change the future" from Shinsekai Yori is also really good. Exactly what it says on the tin, you can't change anything without changing yourself. If you want to make a better future, you have to change the way you act in the present. It's true at both the personal and societal levels.
"You found me" from Anohana. Look, there's no deep or poignant reasoning behind this one. It hits only in the context of the show. If you know, you know.