Translation:He became sad because I cut the dress with scissors.
So, the thing that makes this 'because' is 'için'?
Would "Makas ile elbiseyi kestiğimden o üzüldü" also work?
Yes, replacing "için" with "-den" also works.
In fact, when used to state cause, "-den" is short for "-den dolayı".
The following sentences have the exact same meaning:
Geldiğim için sevindi. Geldiğimden dolayı sevindi. Geldiğimden sevindi.
Omitting "dolayı" can cause some confusion if the reader is not familiar with the noun, as it is the ablative case.
Acıdan bağırdı. (S/he shouted due to the pain.) Balkondan bağırdı. (S/he shouted from the balcony.)
This is not an important issue as you can decide by the context.
"She got sad because of the dress that I cut with scissors" - can someone explain to me why is this one wrong? I'm lost...
If you report "seeing as," I would probably accept it. "as" is a little more blurry, but I would have to think about it and ask around. :)