Here is what varied teachers of the Buddhadharma have to say about the meaning of this seemingly self-evident slogan:
You would like to put people in the wrong by saying disparaging things. However pleasantly coated with sugar and ice cream, underneath you are trying to put people down, trying to get revenge... You think that your virtues can only show because other people's are lessened, because they are less virtuous than you are. - Chogyam Trungpa
The next one is very easy to understand: "Don't malign others." We put a lot of energy and time into gossiping about others. Perhaps there's somebody, maybe it's just one person, that you have a problem with. Maybe it's Pearl, who is so pitiful. She is always feeling left out, and you find yourself reminded of your mother, who's also like that. Somehow Pearl and your mother become all mixed up together, and you find yourself continually irritated and disgusted by the pitifulness of Pearl, and it keeps triggering a lot of stuff in you. Yet you don't have the slightest interest in actually getting to know Pearl and finding out what's going on there. You have no desire to communicate with Pearl and find out who she is. Instead there's some kind of satisfaction that you get from not liking her, and you spend a lot of time and energy talking to yourself about Pitiful Pearl, or whoever it might be - Horrible Horatio or Miserable Mortimer. - Pema Chodron
Here the slogan is translated as "Don't Be Excited by Cutting Remarks":
In general, don't take joy in disparaging others. In particular, when another person says something bad about you, don't respond by talking maliciously about him to others. In fact, even if some injury has resulted, strive always to praise the good qualities of others without blaming this or that person. - Jamgon Kongtrul
Translated as "Do Not Laugh at Malicious Jokes":
The commentary here says more than the verse itself. Do not make bad jokes. The author is not advising us to avoid bad puns, but is referring to malicious sarcasm. Don't make fun of other people in ways that would bring pain to their hearts. The temptation is especially strong when it entails the double satisfaction of disparaging another person and exalting ourselves at the same time by showing off our cleverness. Those of us prone to this type of humor need to address this by changing the conditioning of our speech. All types of harsh speech should be abandoned to avoid harming ourselves and others as well. - Allan Wallace
Translated as "Do Not Meet Abuse with Abuse":
If people say to us, 'You are not a good practitioner. You vows are useless,' we should not respond, by pointing out their defects, for instance telling a blind man that he is blind, or a lame man that he is a cripple. If we act like this, then both parties will be angry. Therefore, let us not utter a word that will harm or make others unhappy. When things are not going well, we should not blame anyone else. - Dilgo Khyentse Rinpoche
Simple and self-evident but oh so hard to put into practice.
The original sources for these commentaries may be found here.