1. What does it mean?
미쳤다 is a form of 미치다, used casually to react to an extreme experience. A great performance, an outrageous price, or a surprising result can all prompt it; the surrounding words decide whether it is praise.
Also written or said as 미쳤네, michyeotda
2. The nuance
It is not a medical description. Calling a person 미쳤다 can be insulting, especially outside friendly joking. Adding 요 changes the speech ending, but does not automatically make the judgment respectful.
3. Examples
“이 작은 스피커 소리 미쳤다. 공연장 같아.”
This tiny speaker sounds incredible. It's like being at a concert hall.
“택시비 보고 미쳤다 싶어서 지하철 탔어.”
The taxi fare looked ridiculous, so I took the subway.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.