ag.algebraic geometry - Extending Functions on Closed Submanifolds of C^n

Yes, this is true. It follows from "Cartan's Theorem B" which says that H^1 of any coherent analytic sheaf on a closed submanifold of C^n is 0; the same result is also true for analytic subspaces. Look up any book on several complex variables for a proof. (It is quite possible that there is a more elementary proof.)

(One uses the theorem as follow: Let X be the submanifold or analytic space and consider the exact sequence of sheaves on C^n

0 --> I --> O_{C^n} --> O_X --> 0

where I is the ideal sheaf of X. The vanishing of H^1(C^n,I) implies that the map H^0(C^,O_{C^n}) to H^0(X,O_X) is surjective, which is what you want.)