Taylor’s theorem for multivariate functions.
\begin{align*}
f(\mathbf{x}+\mathbf{h}) &= f(\mathbf{x})+\mathbf{h}^T \nabla f(\mathbf{x})+\dfrac{1}{2!}\mathbf{h}^TD^{(2)}f(\mathbf{x})\mathbf{h}+\dots
\end{align*}
where $ \nabla f(\mathbf{x}) $ is the vector of first partial derivatives of $ f $, and $ D^{(2)}f $ is the Hessian matrix of second partial derivatives. Applying this to $ F(\mathbf{W}_t, t) $, we find
\begin{align*}
F(\mathbf{W}_{t+dt},t+dt) &= F(\mathbf{W}_t+d\mathbf{W}_t,t+dt)\\
&=F(\mathbf{W}_t,t) +d\mathbf{W}^T_t \nabla F(\mathbf{W}_t,t) +dtF_t(\mathbf{W}_t,t)+\dfrac{1}{2!}d\mathbf{W}^T_tD^{(2)}F(\mathbf{W}_t,t)d\mathbf{W}_t.\end{align*}
But
\begin{align*}
d\mathbf{W}^T_tD^{(2)}F(\mathbf{W}_t,t)d\mathbf{W}_t &= \sum_{j=1}^{n} \sum_{k=1}^{n} M_{jk} dW_{j,t}dW_{k,t} \dfrac{\partial^2F}{\partial x_j \partial x_k}(\mathbf{W}_t,t)\\
&= dt \sum_{j=1}^{n} \sum_{k=1}^{n} M_{jk} \dfrac{\partial^2F}{\partial x_j \partial x_k}(\mathbf{W}_t,t)\end{align*}
hence
\begin{align*}
F(\mathbf{W}_t+d\mathbf{W}_t,t+dt) &=\\
F(\mathbf{W}_t,t)&+d\mathbf{W}^T_t \nabla F(\mathbf{W}_t,t) +dtF_t(\mathbf{W}_t,t)+ \dfrac{1}{2} dt \sum_{j=1}^{n} \sum_{k=1}^{n} M_{jk} \dfrac{\partial^2F}{\partial x_j \partial x_k}(\mathbf{W}_t,t)\end{align*}
subtract $ F \mathbf{W}_t,t $ from both sides to obtain $ dF (\mathbf{W}_t,t) $ }
\begin{align*}
dF (\mathbf{W}_t,t) &= d\mathbf{W}^T_t \nabla F(\mathbf{W}_t,t) +dtF_t(\mathbf{W}_t,t)+ \dfrac{1}{2} dt \sum_{j=1}^{n} \sum_{k=1}^{n} M_{jk} \dfrac{\partial^2F}{\partial x_j \partial x_k}(\mathbf{W}_t,t)\end{align*}
grouping the $ dt $ terms together
\begin{align*}
dF (\mathbf{W}_t,t) & = d\mathbf{W}^T_t \nabla F(\mathbf{W}_t,t) +dt \biggl(\dfrac{\partial F}{\partial t}(\mathbf{W}_t,t)+ \dfrac{1}{2} \sum_{j=1}^{n} \sum_{k=1}^{n} M_{jk} \dfrac{\partial^2F}{\partial x_j \partial x_k}(\mathbf{W}_t,t)\biggr). \\ \textbf{QED}
\end{align*}
When $ M $ is the identity matrix…
$$d\mathbf{W}_td\mathbf{W}^T_t = \mathbf{I}_ndt$$
which is the multivariate form of Ito’s Rule. This can otherwise be written in different notation for individual components as
$$dW_{j,t}dW_{k,t} = dt\delta_{j,k}$$