Let's suppose you spin the barrel before pulling the trigger of the revolver. Clearly, there is a probability $$ P(\text{spin}) = \frac{1}{3} $$ of a bullet being fired. What about if you pull the trigger without spinning the barrel? Let's consider the chambers I could have pulled the trigger on when it was my turn. There is a single chamber which I could have had which leads to a bullet being in the chamber on your turn, with the other three chambers leading to empty chambers on your turn (draw out the chambers!). As a result, the probability of a bullet being fired if you don't spin the barrel is $$ P(\text{no spin}) = \frac{1}{4}. $$ Therefore you should not spin the barrel before pulling the trigger.