Why is fractional reserve banking allowed? I learned about banks' fractional reserves and that revolted me. How is that not considered to be fraud? 

The idea doesn't seem too sophisticated, on the contrary, it sounds very simple: the bank just loans money they don't even have and charges an interest rate (which is not low at all) on that virtual money. When I go to a bank to take out a loan, they will just type the amount of money in my account (but the money does not exist) and I will introduce the money into the economy (when I use that money to buy something). 

Why are fractional reserves not considered fraud?
I recommend a careful read of the Wikipedia page for Fractional-reserve banking because it seems you are confused about how fractional reserve banking works.
Banks only lend out money they have under fractional reserve banking and, in fact, strictly less than they have. Those funds are a mix of deposits, funds from other debt issuance, and bank equity. They don't lend out funds they don't have and they don't simply credit accounts at will in the way you suggest. When they credit a borrower's account with funds, they have to debit those funds from elsewhere on their balance sheet. So, lend-able funds go down as borrower's funds go up, and then the resulting loan is a new asset of the firm. While they do create money this way, they do so at the expense of these lone-able funds so the loaning out doesn't create new assets for the bank.