https://dict.leo.org/englisch-deutsch/why I didn't see an actual question, so I'm not sure how to answer? Maybe: don't use google translate so much! Or: always put context around words, translating them individually rarely renders desirable results. Or: what?

What's the question? (Apparently the video didn't answer it?)

You should definitely wait for a native speaker to answer, so I'm just leaving my two cents as a comment! In my personal experience, I've only used the first three to mean "why". "Warum" and "wieso" are I think basically the same (I realised my German friends use "warum", Swiss friends use "wieso"). "Weshalb" also seems to me to be a synonym for the first two, but less common. I personally use "darum" only to ANSWER "Warum?" or to mean something like "of that". It's in the same "type" of word for me as darauf, daran, etc. I've never used it as "why". I think sometimes you can use it in a similar way to "deshalb" and "deswegen" but I'm not sure. "Deshalb" to me clearly means "therefore", as in "I am sick, therefore I couldn't go to school". Also never used it as "why". I use "deswegen" when I want to say "that's why", as in "He was lazy, and that's why his business failed". Actually it's kinda like deshalb? But I have no idea how native speakers make the distinction.