Well, first of all there are obviously legitimate reasons to take offence to something. Shaming people for genuine transgressive acts can serve as a check and balance to society. The problem is when it is not because of a person's actions or the content of their argument but rather the language they use. So an example would be Tim Hunt who joked about women in science labs. The evidence suggests it was just a joke, but because of the subject of the joke and the language used it was deemed 'offensive' and lost his job.
Consider also the UN report on Cyber VAWG(Violence Against Women and Girls). Their argument for tackling the issue was to ask governments to create legislation that effectively creates a dangerous precedent for Free Speech online. Now harassment online whether against men or women is a new issue we face, but what we need is a fair and measured approach to this. Anita Sarkeesian for instance, when speaking at the UN not only spoke about real threats, but being called a 'liar' or that she 'sucked' was also an issue. Not only that but videos on Youtube that criticised her views (what she called 'hate' videos) was an issue. Now regardless of whether you agree with the criticism hurled at her on those youtube videos or not, most of them are still just criticism or ridicule. There is huge difference between videos like that and threats.
Not only that, stuff about online pornography and violent video games was snuck into the report. This has nothing to do with the issue at all.
The truth is, some people are always going to be easily offended, the problem is when there are real world repercussions. So in the case of Tim Hunt he lost his job. That creates a chilling climate where people no longer feel free to be able to speak their mind no matter how reasonable their argument is.
I am an advocate for unadulterated freedom of speech because I believe that bad ideas will be chased out within an open and free expression. I think banning ideas will make the problem worse because those ideas are never challenged. It also means that legitimate ideas that are true, but just happen to be extremely controversial are never aired. In that case it would be a hindrance to progress.