When I feel remorse, it's basically that part of me feels like I've stepped into the shoes of the person I hurt, and I'm feeling their pain and thinking about how they'd see me, and meanwhile I'm comparing all that with what kind of person I'd like to be and seeing that this particular action falls short of it.
For a specific example - I volunteer with disabled kids. Once, I was working with an autistic girl, taking her swimming as part of an activity program. For the first two days (in two separate pools) she swam happily, but the third day she refused to get in the water. I felt sure that as soon as she got in she'd enjoy it, so I was pushing her pretty hard. Finally I suggested to the other volunteer that we just pick her up and toss her into the water, and the other volunteer said she thought that would be wrong. So I gave up on getting the girl in the water and just let her draw on some paper instead.
Later, thinking back on that, I was horrified by my suggestion. I thought about how I'd have felt about a teacher forcing me into the water after I'd made it clear I didn't want to go in, how scared and angry and helpless I'd feel, especially if (like this girl) I didn't have the verbal skills to express my feelings. I thought about the teacher who have done things to me against my will (like dragging me out of hiding spots) and how I felt about that. And I thought about what kind of teacher I wanted to be and how I wanted the kids I worked with to feel about me and about themselves, and what I'd contemplated wouldn't fit with that at all.
But it's not just thoughts. All of this brings up intense emotions. I feel disgusted by myself, I feel ashamed, I feel sad and mixed up inside, etc. The day afterwards I felt out of sorts all day, until I'd confessed to my parents about this and decided that next time I'd apologize to the autistic girl (unfortunately her parents pulled her from the program because they'd mostly expected her to enjoy the swimming and she wasn't enjoying it).
It's hard to describe, I know exactly how it feels but I have no idea how to convey that to someone who doesn't feel that particular feeling. Kind of like when others try to describe sexual desire to me (I'm asexual). I really don't get how sexuality feels, I just get enough to realize it's something I don't feel. I'm guessing it's similar for a psychopath with regards to understanding remorse.
Also, with regards to things I can't help, I don't feel remorse about those. (Some other people do, but not me.) For example I have PTSD and mild autism and both of those combine to cause meltdowns when I'm upset or overwhelmed. Something will set me off and I'll start screaming, accusing my parents of not caring about me, threatening to hurt myself, sometimes even shoving or hitting my parents. I don't feel remorse for doing this. I wish it didn't happen, but I have no control over having meltdowns and given that I'm having a meltdown I generally choose the best course of action I was capable of. I know it hurts my parents when I do this and I'd like to figure out ways to avoid doing it for their sake, but I don't feel like a bad person for doing those things and I don't feel the strong compulsion to make up for it in some way. (Mostly I just want to forget about it because my meltdowns are far more unpleasant for me than for others.)
I hope this clarifies things somewhat. I'm not exactly normal in my emotional experience, but I'm certainly not a psychopath. (My guess is I'm actually less psychopathic than most people, because I don't seem to feel enduring hatred even for people who've hurt me deeply, such as the ones who caused my PTSD. This trait is probably partly why I'm trying to understand psychopaths instead of just calling them bad.)
Oh, and lastly - why I'd do something I consider immoral? Because I'm not perfect and I don't always think through the implications of what I'm doing at the moment. It's not always obvious as I'm doing something that it's something bad, it's only on reflection later that I realize it.