Can this phrase be used in the meaning "nothing personal" like in situations when you have to reject someone's request for example?
how can I understand that the double negation..."nothing"... "not" ---turns into " is personal"??
The Hebrew here does not have a double negation; שום דבר is a "positive" word that technically means "something" (although most people don't make this distinction in Hebrew anymore). Here is an explanation in Hebrew: http://hebrew-academy.org.il/2012/08/08/%D7%9B%D7%9C%D7%95%D7%9D-%D7%95%D7%A9%D7%95%D7%9D-%D7%93%D7%91%D7%A8/
In any case, even if there was a double negative here, the answer to your question has more to do with how negatives work in English. This is why a double negative in Hebrew (or many other languages) can be translated into a single negative in English.