If it doesn’t seem plausible that your emotions, which exist in your head, are experienced as bodily sensations, consider this: When someone is experiencing and acting on a strong emotion, they look different, right? An angry person holds themselves differently than a non-angry person—more aggressively, you can see they’ve tensed up, they may have a different facial expression, etc.
That way that they look different? It feels different from the inside. If you’ve tensed up, you are literally tense and you can feel that in the pattern of muscular activation. That’s your body preparing for a fight.
There is a lot going on in your brain that isn’t easily directly consciously activated. This isn’t mysterious and mystical, it’s just obvious and mundane. Consider again catching a ball—you don’t consciously think through the process of catching the ball, you just catch it. Similarly walking, or riding a bike, or brushing your teeth. Most actions even where you consciously intend them you’re at least partially handing over to some unconscious part of your brain.
Among those unconscious processes are things responsible for a lot of your emotional responses, and one of the things they do is get your body ready for the action. If part of you decides you need to gear up for a fight (getting angry) it gets your body ready to gear up for a fight.
I also liked this on feelings being on the body: