These are wise words from my dad, though I will admit they are easier said than done (i.e. I pretty much never follow this adage). I definitely have a hard time letting things go, and I can hold a grudge for a long time (not something I'm proud of). But this a good saying to have in the back of your mind when someone says or does something that bothers you. Usually this refers to something minor (ex. someone criticizing your outfit or poo-pooing your favorite restaurant), and that's why you should let it go. I mean, is it REALLY that big of a deal? No. And do you really care if that person has different tastes than you? No (I'll wear/eat what I want!).
It's not so much as what people say that annoys me, but more about their negative attitudes. In general, I try to be positive, say nice things (at least to your face), and be happy about life. It's difficult for me to be around people who complain all the time or say nasty things, from moaning about the humidity in Maryland during the summer (Um, don't like it? Move.) to gossiping about random people (If I don't know them, I don't care...So why are you talking about them to me?). Life is too short to start off your day being rude or thinking mean things; then you ruin someone else's day, too!
So, try not to lose your groove because of those Negative Nancies or Debbie Downers (why are they all female names? A coincidence? I think not). Keep a smile on your face, and take the high road by not saying anything snarky back (but think it all you like. If people could read minds, no one would like me).