The current issue seems quite clear to me and is summed up well by Lord Dear, former chief constable of West Midlands Police, and former shadow home secretary David Davis.
Quoting from the article linked in the OP (my bold):
Mr Davis said: "The simple truth is that in a free society, there is no right not to be offended. For centuries, freedom of speech has been a vital part of British life, and repealing this law will reinstate that right."
What is offensive, or potentially offensive, to one person may not be so to someone else. For instance, on a very mild level, some gransnetters have said they object to being called 'dear' because, presumably, in some way they find it offensively patronising. Some others don't find being called 'dear' in the least offensive and in fact rather like it. If one just extends that range of reactions to other, more forceful expressions, or when there is talk on subjects where people have strong feelings (religion, politics, etc) it's easy to see how muddled things could quickly become.
Censorship is always dangerous. Preventing discrimination is another issue altogether.