The equality $\frac{\mathbb{P}(Y\mid X)}{\mathbb{P}(Y)}=\frac{\mathbb{P}(X\mid Y)}{\mathbb{P}(X)}$ means that in a supermarket analysis, knowing that a customer bought milk ($X$) multiplies the probability of him buying cookies ($Y$) by the same factor that knowing that a customer bought cookies multiplies the probability of him buying milk.
In association rules theory, this says that the lift of a rule is symmetric.
I've tried looking at specific cases to generate intuition: when one of the events has probability 1 the fractions are both 1, and when X always implies Y this is somewhat logical since knowing that Y occured allows us to to only look inside Y instead of at the whole space of events, hence multiplying the chance of X happening by $\frac{1}{\mathbb{P}(y)}$.
In general, why should it be true?
Suppose people who buy milk are more likely to buy cookies than people who do not buy milk are to buy cookies.
The claim is that this implies that people who buy cookies are more likely to buy milk than people who do not buy cookies are to buy milk.
To me this seems intuitively reasonable: the initial statement suggests to me that milk and cookies purchases often go together more than you might expect if they were independent, while the second is essentially the same point that cookies and milk purchases often go together more than you might expect if they were independent.
In rhetoric this can be pushed too far: for example people who run $100$ metres in under $10$ seconds are more likely to be men than to be women, but this does not imply that I am more likely to run $100$ metres in under $10$ seconds than my wife is. We are both far too slow.