Let $$\pi:E\longrightarrow M$$ be a complex vector bundle with an hermitian metric $h$. I am using Griffiths and Harris' Principles of Algebraic Geometry, where it defines a connection on $E$ as a map $$D:A^0(E)\longrightarrow A^1(E)$$ such that $$D(f\sigma+\tau)=df\otimes\sigma+fD\sigma+D\tau$$ and it says that the connection is compatible with the metric if $$dh(\sigma,\tau)=h(D\sigma,\tau)+h(\sigma,D\tau)$$
I don't understand what the rhs means, since $$\tau\in A^0(E),\quad D\sigma\in A^1(E)\Rightarrow \tau(x)\in E_x$$ but $$(D\sigma)(x):T_xM\longrightarrow E_x$$ so how do apply $h$ to that pair?
Well $dh(\sigma,\tau)$ is a one-form that eats tangent vectors. So is the right-hand side. So it means
$$dh(\sigma,\tau)(v)=h(D_v\sigma,\tau)+h(\sigma,D_v\tau).$$
Now the arguments of the metric on the right-hand side are sections of $E$, and everything makes sense.