Understanding the chain rule in probability theory

$$p(x,y|z) = \frac{p(x,y,z)}{p(z)} = \frac{p(x|y,z)p(y,z)}{p(z)} = p(x|y,z)p(y|z)$$

On the first step we use the definition of conditional probability. On the second step we use the same definition on the numerator to convert the joint probability $p(x,y,z)$ into a conditional $p(x|y,z)$ and a joint $p(y,z)$. Finally, we divide $p(y,z)$ by $p(z)$ applying once again the definition of conditional probability, and we obtain the result.

Another way of looking at it is that you can just ignore variables that are always on the right side of the conditional sign. In that case the expression is just the usual conditional probability:

$$p(x,y) = p(x|y)p(y)$$

You simply condition all of these probabilities on $z$ and you get your original formula.