SPOILERS YOU HAVE BEEN WARNED IF YOU SOMEHOW HAVE MISSED OUT.
Beautifully written article from the AV Club explaining just why the latest GoT episode is so affecting and why we care so much.
I think the author hits two keys points right on the head. First, as she puts it, this moment represents "the death of hope". That expression gave me chills, because it perfectly summed up what I felt when I read the books for the first time. Robb was the last hope for the audience in terms of a quality ruler who could take over Westeros in the short term (since Dany is perpetually busy over the sea). With his death, the idea that a worthy leader is going to roll in and save everyone pretty much falls away.
Second, and I think the more important point, is that it further emphasizes the "no one is safe" idea. It's not so much that we care about Robb and Catelyn - it's that their deaths mean that the characters we REALLY care about (and up to this point in the show, I can guess that would be Tyrion, Jon, Dany and Arya) are not safe. The idea of watching poor little Arya die is a thought I can barely stand to entertain. But now it's like a sword hanging over the head of every character I truly love, and that creates an unbearable and yet completely compelling tension that makes me push through even the weaker parts of the narrative.
I'll open this up to comments, but I ask that you clearly mark spoilers so people who are only watching the show aren't spoiled on upcoming parts of the books. I'd say even talking in vague terms about what a certain character might do in book 6 or 7 is uncool, because then you know that character actually makes it to the end of book 5.