I have that problem too. I tend to keep quiet because people seem to get offended so easily, but then I also have a short fuse and get angry about some things very quickly and spew all over people. So for me it's kind of all or nothing.
As far as I can tell this is what happened with the two situations you describe:
persian85033 wrote:
Just on Friday, me and a coworker, well, there were a bunch of other people having a discussion. It was about that new law that's been passed here in Arizona.

And I comment to him, 'they were having quite an interesting discussion there'. He replies 'I know, that's why I left before I exploded. Sometimes it's best to keep your mouth shut'. I couldn't quite help it, but said 'I just don't understand the whole thing. It seems ridiculous', and he replies 'sometimes it's better to keep your mouth shut'.
The new Arizona law is a real hot-button issue and many people are very very upset about it. When your coworker said he didn't want to talk about it because he might explode, and you gave your opinion about the issue itself, he was letting you know he wasn't going to engage in a discussion about it because he was already upset and didn't want to explode. I don't think he meant he wanted you to shut up, it was more that he wanted
himself to shut up.
persian85033 wrote:
The same thing happened some time ago, with the lady at the coffee cart in the lobby. That was because they were removing all the tables from the lobby, and I butted in when they were talking, and said 'maybe they don't want people eating in the lobby. Maybe they think it gives an unproffessional image or something'.
Here, it sounds like she thought you were trying to criticize her business, by saying, or so she thought you were saying, that they were making the lobby look sloppy.
I dunno, just trying to make sense of it.