I only have an answer when $M'$ is the terminal multicategory. This can be helpful however for the general solution too. See the considerations below.
The left Kan extension when $M'$ is the terminal multicategory $1$ (this has one object, and one multimorphism for each $n$) can be constructed as follows.
A functor $1 \to \mathbf{Sets}$ is just a monoid $A$ in $\mathbf{Sets}$. The 2-cell which will exhibit it as the left extension of $P : M \to \mathbf{Sets}$ along the unique $M \to 1$ consists of maps
$$d_X : P(X) \to A,$$
for each object $X$ of $M$, such that for any multimorphism $f\colon (X_1,...X_n) \to X$
$$P(X_1)\times...\times P(X_n) \xrightarrow{P(f)} P(X) \xrightarrow{d_x} A$$
$$=$$
$$P(X_1)\times...\times P(X_n) \xrightarrow{d_{x_1}\times...\times d_{x_n}} A\times... \times A \xrightarrow{\text{mult.}} A.$$
Given $P$ one constructs $A$ as follows. Take a coproduct $\amalg_X P(X)$. Take the free monoid on this $\mathcal{Fr}(\amalg_X P(X))$, and factor this out by the relations
$$P(f)(x_1, x_2,\ldots, x_n)\sim x_1x_2...x_n.$$
The $d_X$ will be defined as the canonical maps.
For the general case, one could follow Kelly, who defines point-wise Kan extensions withing a general bicategory with comma objects. (I'll insert a reference for this.) Using this for the 2-category of categories, the usual formula for $Lan_F(P)(X)$ is obtained by finding the left Kan extension of the composite
$$P\downarrow X \to M \to Sets$$
along the functor $P\downarrow X \to 1$ to the terminal category. This happens to be just the colimit. So for categories, finding point-wise formulas are reduced to finding Kan extension along the functors to the terminal category. This however relies one the fact that an object $X$ of a category can be given by a functor $1 \to M'$.
I believe that the 2-category of multicategories has comma objects. Unfortunately, the multifunctors $1 \to M'$ are monoids in $M'$ rather than objects.
