How we change what others think, feel, believe and do
Apologize before asking for something or giving information that might otherwise cause the person to be upset or indignant.
The basic structure of pre-apology is 'I'm sorry ... but ...'. You first apologize, then make your request or otherwise tell them something that is not necessarily what they want to hear.
I'm sorry to trouble you, but can you help me move this?
I'm sorry to have to tell you this, but your son has been causing a problem.
I do apologize, but I'm going to have to return these.
As well as reparation for doing something wrong, an apology is an acknowledgement of distress or trouble caused to other people. It also shows active care and empathy, which they may suspect is lacking when your words make them unhappy.
When you show that you know they are feeling upset, then this draws the sting so they do not have to tell you that they are unhappy about what you said. They may still do so, but the pre-apology takes the power and surprise out of their rebuke.
It is important that the pre-apology sounds genuine. If it is said in a throwaway style then it will likely add to anger rather than remove it. The tone of the overall conversation and the character of the person will also be taken into account. People are good at detecting deception and falseness which, if identified, can reverse the benefits of faking.
'But' can mean 'ignore what was just said' although can work well in this setting. If in doubt, you can just leave it out and substitute a brief pause.