The construction as suggested by Loïc Teyssier can be globalized, with the help of a Stein neighborhood basis.
By Corollary 1 in ''Every Stein subvariety admits a Stein neighborhood'' by Siu, every Stein 
manifold $M$, which is a submanifold of some manifold $N$ has a neighborhood $W$ in $N$ such that there exist a holomorphic retraction $r : W \to M$. In particular, when $N$ is Stein (as is the case here, with $N = \mathbb{C}^d$), then also $M$ is Stein, and the theorem applies.

Thus, if $U \subseteq M$ is open, $f : U \to \mathbb{C}$ is holomorphic, and one lets $V$ be the open set $V := r^{-1}(U) \subseteq W \subseteq \mathbb{C}^d$, then $F = f \circ r : V \to \mathbb{C}$ is an extension of $f$.