Countably many dense subsets in forcing

Question

I am reading some introductions to forcing such as this one https://arxiv.org/pdf/0712.2279.pdf and they seem to take for granted that the family of dense subsets of a countable, separative, atomless partial order is countable (see page 9 in the link). Maybe this is a trivial combinatorial fact but I can't see it. For sure, that doesn't hold in some non-separative posets such as $\mathbb{Z}$.