It seems to be folklore that if we have an actegory, i.e. a monoidal functor from a monoidal category $C$ to an endofunctor category $Cat(D,D)$, we can obtain from it a monad on $D$. This appears for example here and here. Unfortunately I can't seem to find a reference, or a detailed description of how this monad is actually defined. (Intuitively I would expect a sort of colimit with some coherence conditions.)
Can someone tell me where I can find it, or at least "why it is obvious"?
I'm going over what's already in the comments a little here, but: given an action of $C$ on $D$ as described, any monoid in $C$ gives rise to a monad on $D$. This is because a monoid in $C$ is just a (lax) monoidal functor $1\to C$, and composing it with the functor $C \to Cat(D, D)$ given by the action then we get a monoidal functor $1 \to Cat(D, D)$; i.e., a monoid in $D$.
When an author talks about 'the' monad obtained from an action, almost always they are referring to the monad so induced by the monoidal identity $I$ in $C$, together with its natural monoid structure $I \otimes I \to I$, $I \to I$. See, for example, page 12 of these notes (where the author uses the terminology $J$-monad to refer to an action of the monoidal category $J$).
However, it's worth pointing out that neither of the examples you have given are doing this: in both cases, they give some fixed actegory $C \to Cat[D, D]$ and then show how you can turn monoids in $C$ into monads on $D$. The operads article exhibits an action of the monoidal category $Psh(\mathbb P)$ upon $Set$; since an operad is just a monoid in $Psh(\mathbb P)$, we therefore get a monad in $Set$ from each operad.
If we did want to talk about 'the' monad generated from the actegory in this case, it would be the one induced by the operad given by the natural monoid structure on the identity in $Psh(\mathbb P)$, which is the commutative monoid operad $Comm$ and gives rise to the finite formal sum monad on $Set$.
Similarly, the article on clubs gives us an example of a ($2$-)action of $Cat/\mathbb P$ on $Cat$. A club, being a monoid in $Cat/\mathbb P$ will therefore induce, through this action, a ($2$-)monad on $Cat$.
In this case, 'the' monad induced from the action itself is the rather boring identity action on $Cat$, which is iduced from the natural monoid structure on the identity $(1, 1\mapsto \{*\})$ in $Cat/\mathbb P$.
