How to explain high sensory processing sensitivity to (own) child?

For children in this age it is not possible to understand the theory of SPS (or other complex things). Try to find a simple picture, that represents the most of your trouble. For your case I could imagine something like an invisible shield or fur every person has. Something that protects persons against loud noises, powerfull emotions and so on (Use the things, you have to struggle with). Then you could talk with them about the thickness, or the seize of this special protection, and that no one has the same like others.

Now you could explain, that your shield is not as large, or that you sometimes need time to find it. Maybe you could explain your reactions at triggers like "fighting" the emotions because the shield is missing. Children know strong feelings and that one could be overwhelmed by them. You could use any symbols/pictures you feel good with and your children know from children-stories. (The shield for example from knights tales.)

As your children grow older they will ask more detailed questions and you could give more detailed answers. If they ask, you could try the simplest answer and let them ask again, if it is not enough information.

I am also sensitive to certain stimuli, such as repetitive noises (which kids are prone to) and smells.

In the case of repetitive noises, after the first couple, when it's obvious the kid is entranced by what they're doing, I am able to say (calmly but very firmly) "Mom can't stand repetitive noises, you need to stop right now." My kids always complied, even my autistic kid who has a hard time putting herself in other people's shoes. Kids can and do care about their parents' well-being. That being said, they do forget in about 30 minutes and will always forget until they are older. Be patient and tell them to stop as many times as it takes--always calmly and firmly. My kids are 18 and 12 now and I don't remember the last time I had to ask them to stop.

I found that being straightforward and specific with my needs was enough. I didn't have to explain why. They never asked, they just accepted it. Don't feel compelled to explain to them beyond "it really bothers me" or maybe "makes me very unhappy." Do acknowledge they're doing something normal for kids to do, but that it's something they can't do around you as a matter of compassion and politeness.

It sounds like you want to explain to them partly because you feel guilty. So--frame challenge--how do you stop raising your voice and behaving angrily when you're so sensitive? One, know your triggers and act as soon as you realize it's happening. Do not wait until you can't stand it anymore. Two, know your "levers"--things that you can do to return to a normal state. It sounds like you know how to take a time-out, just take it before you get upset (but yes, you have every right to ask them to stop what they're doing, even if it's normal kid stuff; they'll have plenty of opportunities to do whatever it is and it's okay to teach them compassion by having them stop something they're enjoying because it hurts someone else).

I also have challenging traits I've passed along to the next generation! What I've found helpful in communicating with my children about those is to frame it less as helping them understand my issues --why would they care? --and more showing them that I understand THEIR issues (because I share them). It's the same end result, but it starts from where they are, not where I am.

For instance, I've always struggled with insomnia. My parents never really seemed to understand it when I was a kid, but when I see my son experience the same problem, I know exactly what to tell him. Similarly, both of us require a large amount of physical activity for optimum mental health. I can always tell when he hasn't had enough. It took me years to discover the connection, but I've been able to share my coping mechanisms with him from an early age.

Occasionally, there will be something --for instance, if I lose my temper, or am grumpy --that isn't directly related to something they are also experiencing at that moment. In that case, I just apologize, and explain that even grown-ups have to work hard sometimes to do the right thing. Going much further than that, I think, would just teach them to make excuses.