The product axiom for the Giry monad is given as follows:
$$ \mu_{X}: P(P(X)) \to P(X) $$
given by
$$ \mu_X (M)(A) := \int_{P(X)} \tau(A) M(d\tau). \,. $$
$P(X)$ is equipped with the weakest topology which makes the integration map $\tau \mapsto \int_{X}f d\tau$ continuous function for any $f$, a bounded, continuous, real function on $X$.
The list monad is easy to understand because we have simple explanations like "a list of lists goes to a list by concatenation". Can someone give a nice explanation of the product for the Giry Monad in the same spirit as that for the List monad?