Translation:I'm sorry, this is my fault.
so i speak chinese and we would probably say " excuse me, this is my fault" does anybody think so as well?
That is an unnatural literal translation. In English it is much more common and acceptable to say "sorry, my bad" or "sorry, my mistake"
不好意思 is one of those phrases that has no exact English translation. It is a little stronger than 对不起 and is often used to imply that the one who has spoken it has lost face a little. It's typically used in occasions when you might say "I'm sorry" in English.
Doesn't 不好意思 literally mean "it's a bad thought" or something like that? What is the exact relation between this and apologizing?
I would literally translate this to: "Not good meaning". And I think of it as, "Sorry, I did not mean for that 'not good' thing to happen".