The OR represents the odds that an outcome will occur given a particular exposure, this is very analogous to the concept of conditional probability except that the formula for conditional probability makes a much better connection with its definition, why do we use the odd ratios whatsoever when we can just find $P(A|B)$ and then use $$\frac{p}{1-p}$$
to find the odds that the outcome A will happen given that B is guaranteed to happen.