If you have to defend your position or stance on something by prefacing or amending it with “I’m sorry, but that’s my opinion”, you’re doing it wrong.
For one, when you apologise for your position, you’re admitting that there’s a problem with it. If you’re not doing that, then you’re admitting that it’s an unpopular opinion and you’re possibly offending the person you’re expressing your opinion to. So realistically, all you’re telling me is that a) you don’t know why you believe a certain way, or b) that you’re basing your opinion on an irrational prejudice or fear
Unless you can argue cohesively that you’re not a raging uneducated asshole who believes everything they read in the Herald Sun and that there’s a brain underneath all of that organic shampoo which you’ve bought out of guilt, then you probably shouldn’t have that opinion in the first place.
Several examples of this being used in practice that I have seen or heard around:
“I don’t like people from [country/race/religion]. I’m sorry, but that’s my opinion.”
“Gay people gross me out. I’m sorry, but that’s my opinion.”
“I’m one of the rude customers when things like that happen. I’m sorry, but that’s my opinion.”
You might as well preface your statements with “I’m not racist/homophobic/sexist/a bad person, but…” and just save me the trouble of listening to you so I can think about happier things. Like awkward hugs between black and white people, and hilarious screaming Cantonese girls fighting over a guy.
So, in conclusion, unless you’re actively trying to acknowledge that your opinion sucks, please do not apologise for your opinion. It just doesn’t make sense.