A problem I sometimes have is that someone will suggest that I do something uncomfortable - physically or emotionally, whichever - and they'll acknowledge that it's not going to feel good.
And then they won't follow up by explaining exactly how much and in what way it's supposed to hurt, let alone why or how to mitigate that.
Look: if you're going for a run, you might get a stitch in your side, your legs might cramp, your lungs might burn. If you feel a sudden stabbing pain in your foot, that's not the run. You've stepped on something sharp and you need to stop and call for a ride, not tough it out. If you've been told that running might hurt and that's a good thing, a sign of weakness leaving the body or whatever, and there's any ambiguity between "stepped on something sharp" and "this is how it's supposed to hurt" - are you supposed to have blisters or do you need better-fitting shoes? Are you supposed to have a headache or do you need to drink way more water than you have been? Is a sunburn just part of a healthy experience that means you're making vitamin D, or should you be wearing sunscreen next time? - then you're blundering into a problem that you would have noticed and solved if you had encountered it by sitting on the couch.
If you're apologizing for something you've done wrong, you might feel guilty, stared-at, awkward, resentful. If you feel like the walls are closing in and you're going to puke, then you have a more serious problem and might want to get that looked into, not chalk it up to "well, acknowledging that you've made a mistake can be painful". If you're apologizing to someone who's abusing you, then how much it hurts is a red flag. If you're in a badly facilitated therapeutic or moderated conversational environment, and you don't know how much it's supposed to hurt, then when it hurts more than it's supposed to you won't notice that you need to fix or leave the setup. If you're apologizing for something that isn't wrong, because you've been indoctrinated in a normset that's unhealthy about what things are violations, then by letting that normset also tell you that it's supposed to feel bad you're losing a signal that could tell you to bail.
Even things you aren't actively doing can have this issue: it took me a long time to be really sure that most people don't experience pain when they have their blood pressure taken, because every time I complained about it, an indifferent nurse would say "yep, lots of pressure, mm-hm, it squeezes pretty tight" and not "actually, while you're unlikely to be in medical danger, that is not typical; we might make different tradeoffs about how often to recommend this test if it hurt everyone like it does you".*
And if you don't know how something is supposed to hurt, and know that you don't know it, you will need some risk-aversion there to avoid blundering into sharp objects and mental health triggers and cost-benefit analyses that were not designed for how you're put together. So this can cut off opportunities to try things, the opposite problem from trying too hard at something that isn't working.
If you are recommending that people do something that might be uncomfortable or painful, tell them what normal tolerances are, and what the things that might be causing abnormal responses might be. Demand this of people telling you to do uncomfortable or painful things too. Pain responses have a purpose; ignoring them outright for the duration of an activity which flirts with the damage that aversion is meant to prevent is insane.
*If you have this problem and might give birth and might have an epidural while you do, have them put the blood pressure cuff on your numb leg. Hat tip to Swimmer963.
It's not so much "how much will this hurt" as it is "how much should this hurt". In other words, "how much does it have to hurt before I reconsider". In the running case, for example, you can't know before their run if they'll experience mild muscle soreness of if they'll step on a nail. You want them to know that if it feels like they've stepped on the nail, this isn't what you're talking about, and you shouldn't try to run through that.
There is a distinction between "this is how intense the sensations might be" and "this is the thing they signify, and how bad it is". A lot of the subjective experience of "pain" has to do with the meaning attached to it, and the reaction to that meaning.
In jiu jitsu for example, beginners are often not taught heel hooks in part because the sensation of a knee ligament about to rupture doesn't always stand out as a big deal, and so people will sometimes hurt themselves because they don't notice the warning signs. At the same time, you can get people screaming in pain once their foot is turned the wrong way because all of a sudden the meaning has changed and they no longer feel "okay". Other people can have the same thing happen to them and just kinda look at it like "oops, I screwed that up" because they simply aren't overwhelmed by the idea that their ligaments just tore and their limb isn't pointing the right way anymore.
When you're talking to someone who is in pain (or needs to do something which will be painful), there's two things you want to communicate. One is that it's okay, even though whatever the bad thing is that happened, and the other is what the bad thing is. When you can do those two things, their entire experience can change dramatically.
The same principles apply to emotional concerns. For example, if someone is going to feel embarrassed by something to a degree which seems appropriate and okay, then all you are going to need to communicate is "Yes, this is going to be embarrassing. It's okay". If they're going to be way more embarrassed than is called for (in your opinion), then you *also* want to be communicating that the damage isn't as severe (or call for such an extreme aversion) as it seems. It's the "this sensation means your knee is about to explode" training in reverse. In this case, you aren't just saying "embarrassment is okay", you're also saying "it's not even that embarrassing". Be prepared for people to not just take your word on this, of course, but that is the point of contention.
One way to deal with it is to actually paint them a picture of what it's like from your perspective so that they can see that it's not that big a deal (if they find your story convincing). Another is to just show awareness that it seems super horrible and worth being embarrassed over, and that you don't expect them to be convinced, but that you actually don't think it's that big a deal, for what it's worth. If your opinion means something to them, this can still have a significant effect.
As I read it, the advice Alicorn is giving here relates to the part where you don't want to miss the fact that someone might perceive an embarrassing event as way worse than you do (accurately or not) and then tell them "you should put up with it so that you can do X" without noticing that you might be asking them to endure a much bigger perceived (and maybe real) cost than you actually think it's worth. For example, you might want to say "I'd probably apologize to them if I were in your shoes. And yeah, it's kinda gonna suck. I wouldn't be smiling about it for sure, and I might have a hard time being in a good mood for the rest of the evening, but it's not like it's worth traumatizing yourself over. If it feels like something you can't handle, then that's fine too. It's not the end of the world if this person doesn't get their apology".