The algebras for this monad can be described in essentially the same way: they are sets in which it makes sense to to take "convex combinations" of countably many elements. More precisely, an algebra is a set $X$ together with an operation
$$((a_0, a_1, a_2, \ldots), (x_0, x_1, x_2, \ldots)) \mapsto \sum_{i = 0}^\infty a_i x_i$$
defined for every sequence $(a_0, a_1, a_2, \ldots)$ of non-negative real numbers such that $\sum_{i = 0}^\infty a_i = 1$ and every sequence $(x_0, x_1, x_2, \ldots)$ of elements of $X$, such that
$$\sum_{i = 0}^\infty a_i \left( \sum_{j = 0}^\infty b_{i,j} x_{i,j} \right) = \sum_{k = 0}^\infty \left( a_{p(k)} b_{p(k), q(k)} \right) x_{p(k), q(k)}$$
where $k \mapsto (p(k), q(k))$ is your favourite bijection $\mathbb{N} \to \mathbb{N} \times \mathbb{N}$, and
$$\sum_{i = 0}^\infty a_i x_i = \sum_{j = 0}^\infty b_j y_j$$
whenever $\sum_{i : x_i = z} a_i = \sum_{j : y_j = z} b_j$ for all $z \in X$.

The most concise way of giving the complete definition is to use the monad in your question.