It’s an overwhelming feeling created by a complex set of emotions.
Say you break up with your girlfriend, a few months go by, and you realize you made a mistake.
Now you find out she’s dating your friend.
You’re sad because you realized too late the mistake you made, probably a bit angry at your friend for dating her, and frustrated at yourself for not being more mature in the first place.
This bundle of feelings is what feeling some kind of way is all about.
It’s conceptually very close to the French je ne sais quoi
Which translates to “I don’t know what”.
It means “it’s got me feeling”. Usually you can pick up what “kinda way” they’re feeling from the convo.
I’ve always heard this one used when someone is uncharacteristically grumpy or making odd little rude comments when they usually wouldn’t. Someone else would say they “must be feeling some kind of way” to explain that awkwardness.
Feelings are so subjective that words / descriptions kind of distort the original visceral feeling.