I miss something about offshore

I read here and there that guys got problems with there offshore company to get PayPal, processor , banking, in a simple word to make business.

I have different offshores for my personal use but usually is to make invoices to my regulars companies in Europe. Minimize the profit and secure the money as much as I can.
I use them also to be owners or something where I want to hide my id.

So my noob question is why guys make offshore to make business?

Not speaking about how much business is lost in Europe because not European company but this is another subject.