The Daily Drip
I've had the distinct displeasure of firing employees several times.
From my experience, I can tell you that there are 2 kinds of people who walk out of the HR office after it's all over.
Person 1
They are in disbelief. They can't see what went wrong. They had no idea it was coming and they never saw the writing on the wall. This person could be delusional and may have refused to make the adjustments you recommended, but they also might be a product of you--a crappy manager who didn't communicate expectations clearly or measure performance adequately. If you fire someone and you see this person, you need to evaluate your evaluations and measure your metrics to see where you can improve.
Person 2
As soon as they walked into the room, they knew. You don't even have to say anything. You're usually both sorry. Maybe they regret it and maybe they don't, but the important takeaway from seeing this person is knowing that you were both on the same page. This doesn't mean you couldn't have been a better manager. This is an opportunity to ask what else could have gone better, how you could have better supported them, and assess what was in your control and what wasn't.
In my opinion, the second person is the better option of the two because at least you're both clear on expectations.
Each time I've had to let someone go, I've felt a tremendous sense of guilt and shame. I've always, always, always felt it was my fault. There are some things I couldn't control, but I can't help but think I could have been a better manager for them. I can't help but think there was something I could have done to unlock them, free them, make them feel safe, or to identify that they didn't want to be there in the first place.
It's one of the worst feelings in the world.
When you have to do it, I hope that terribleness motivates you.