Wikipedia states: " In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event"
Conceptually, why is that true? Is there an example of two events being dependent but given a third event they become independent?
Suppose I gave you two coins with unknown but identical bias. In particular, suppose that the probability of $H$ upon tossing either coin is $p \in (0,1)$ with probability $1/2$, and is $q\in (0,1)$ otherwise.
Imagine you tossed each coin once. We then have
$$ \Pr(H,H) = \frac{1}{2}p^2 + \frac{1}{2}q^2 \neq \left(\frac{1}{2}p + \frac{1}{2}q \right)^2 = \Pr(H) \cdot\Pr(H). $$
In other words, the outcomes of tossing both coins are not independent. However, let $A$ be the event that the (common) bias of the coins is $p$. Observe that
$$ \Pr(H,H\vert A) = p^2=\Pr(H\vert A) \cdot \Pr(H\vert A). $$
That is, conditional on the bias of the coins, the two coin tosses are independent, as they should be.