I've always known that phrase, but mostly thanks to Bambi.
I think the movie does a good job of demonstrating the kinds of situations it's really meant for: don't say something mean if it there's no good reason.
There are ways to tell someone something negative while still "being nice." For example, last night my husband and I were out at a lounge with two of his work buddies (railroaders, so 3 "typical guys" and me). I don't know what got into him, but his behaviour was deplorable. I won't get into details, but I was starting to wonder if I'd sat down next to the wrong husband.
We had some words after. I was pissed off and I didn't take the time to calm down and formulate my complaints in a compassionate or constructive way. I got in the car and said "OK, you were acting like a real ass in there" and proceeded to list the things he had said and done.
I'm not happy with how I presented it. Oh, make no mistake, I said what needed to be said. But afterwards, he was completely dejected and felt like a failure as a human being. That was not my intention, and in hindsight, I wish I'd focused on my feelings and his behaviour rather than reverting to juvenile name calling. He puts a lot of stock in what I say, so "You were being an ass because of this and this and this" made him feel like he is
an ass. But he's not, he was just behaving like one.
So the point is... being nice is not always about the content of what you say, but the delivery. There are ways to be "nice" and still say what needs to be said.