an English question

exactly what it sounds like.taking things as they come in small steps day by day.

It means that everyone just took every day as it came. No one knew exactly how each day was going to be after those events occurred, so we all had to take each day one at a time, just living it as best we could. I think it also means that everyone was pretty much forced to be strong and continue life right where it left off, even though there was a lot of suffering and hurt going on in a lot of lives. It was such a very hard time for everyone. I think every heart in America was hurting. But America is strong, and continues to keep moving on ONE DAY AT A TIME.

I would suppose it refers to the people now living their lives more cautiously

not only more cautiosly as the previous poster suggested but also if you take things one day at a time it implys a great level of uncertanty as to what might hapen next.