This quote is total bullshit, and I hate it, and have always hated it.
Usually it's said by parents or teachers, to children. Children. I guess they're hoping kids will somehow learn through this how to shrug words off? How?! The entire way the quote is phrased says it's talking about a current, existing, reality. But seriously, words don't hurt? I would challenge anyone reading to find no time when they have been hurt by words.
Words DO hurt, they can hurt very, very, badly.
That's not to say we should tell our children "the world should be nice to you." Or lead them to believe that the world is a nice place, that you don't need to deal with hurtful words, or that you should let them bring you down for long. The world ISN'T nice, part of growing up is learning to deal with things that hurt, picking yourself up from it and continuing on.
But when someone comes to you, hurt by words, quoting this saying at them trivializes the way they are feeling. It disregards the pain they're feeling, telling them to "suck it up." That's not how you teach them to actually deal with what they're feeling, that's how you get them to bury their pain even deeper inside. And that never ends well, not to mention that it's entirely counter productive to helping them to grow.
Engage the fucking material people. Don't trivialize. Don't minimize. Don't disregard. Engage their pain, do your best to empathize even if those words wouldn't have hurt you, do your best to help them learn to pick themselves back up.