Good question. My daughter who's nearing 2 yrs started up with this, especially with diaper changes and if we pick her up when she's doing something, usually because she needs a diaper change. I think it has to do with wanting to be in control and not having people pester her when "she's doing something important". BUt it's hard to let them continue to play when they've got poop in their pants. So I try to get her to admit to having poop in her pants and needing a new diaper. Then I say, "okay, you still get to play, color, watch your cartoon, or what have you. BUt right now you need a fresh diaper. So fresh diaper, then more playing." Sometimes this works. If not I always grab her legs or arms and hold them to her sides and say, "no hitting or no kicking people". I make her repeat, "No hitting people." and to say sorry to mommy or whoever. I also have her list things that it's okay to hit or kick, like a ball. I think it's one of those things that takes time and consistency. Everything I've read says, they really are too young to control their frustrations at this age, but they do grow out of it. I notice that when I make sure she's gotten a lot of fresh air and energy spent, she usually does better.
There are also books called "Hands are not for Hitting", Feet are not for kicking, Tails are not for pulling, and Teeth are not for Biting. I think it's just about every mom's dread, but typical. Good luck curbing it! If you do find a magic solution, please share it. But my advice as a first time mom, is be consistent with reprimanding the moment the hitting or kicking starts.
My daughter has been highly verbal and speaks in complete sentences since 16 months, so I think it's age and powerstruggle issues, not an ability to say "I'm mad."