When asked to comment, proprietor Feldstein said he had been forced to push the date back twice ( 1.since 2.due to) the weather.
The correct answer is 'due to'.
Could you tell me why?

See a translation

Since implies time has passed. It's more often used with the meaning "from the time that".
If you plug this into the sentence, it doesn't make sense: "from the time the weather." The sentence is not complete here.

"Since" can be used in place of "because", but it's better not to because since has 2 meanings ("from the time" and "because") if the meaning is unclear, it's better not to use the word. If you it into the sentence you get: "because the weather." This also not a complete sentence.

Due to implies a reason for change.
It has the same meaning as "because of" in this case. If you plug it into the sentence: "because of the weather." This phrase completes the sentence, therfore it is correct.

Does this help? I think I can explain it in Korean if you need me to. 필요하면 제가 한국말로 설명 할 수 있을 것 같아요.