In Bell nonlocality, why does $P(ab|xy)neq P(a|x)P(b|y)$ mean the variables are not statistically independent?

The equation

$P(ab|xy) = P(a|x)P(b|y)$

would imply that any dependence that the output $ab$ has on the inputs $xy$ (expressed by the lhs) is solely due to $a$ depending on $x$ alone, and $b$ depending on $y$ alone. This is expressed by the rhs by treating the value of $a$ and its dependence on $x$ as an independent event from the value of $b$ and its dependence on $y$, and hence the probability of a particular $ab$ is the product of these independent probabilities.

In Bell's equalities, we see that this is not the case. Correlations between $a$ and $b$ depend explicitly on which pair of inputs $xy$ is chosen. Specifically, for the CHSH inequality, whether or not $a$ agrees with $b$ depends on both $x$ and $y$, and hence $a$ cannot be said to be independent of $y$ (nor $b$ of $x$). This behaviour means that there will be at least some values of $a$, $b$, $x$ and $y$ where the above equality does not hold.

It perhaps helps to express $P(ab|xy)$ in words:

the probability that Alice gets answer A and Bob gets answer B given that choices x and y were made

Now independence in classical probability holds if and only if $$ P(e_1text{ and }e_2)=P(e_1)P(e_2) $$ where $e_1$ and $e_2$ are events, and practically, you can see what it means through Bayes' theorem $$ P(e_1|e_2)=frac{P(e_1text{ and }e_2)}{P(e_2)}=P(e_1) $$ i.e. for independent events, the condition probability is independent of the conditioning.

Now, we could rewrite $$ P(ab|xy)=P(a|bxy)P(b|xy) $$ you lose a bit of symmetry by doing it like this, but only briefly. Now, the idea of independence in the current context is that Alice's result should not depend on anything that happens on Bob's side, so $P(a|bxy)=P(a|x)$ and Bob's result should be independent of anything that happens on Alice's side, so $P(b|xy)=P(b|x)$. Hence, independence between Alice and Bob would imply $$ P(ab|xy)=P(a|x)P(b|y). $$ So, if this condition holds for all $a,b,x,y$, then the probability distribution is independent, otherwise, the results are not statistically independent.