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We study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal $L^2$-norm. When the tensor power of the line bundle tends to infinity, we obtain an explicit asymptotic expansion of this operator. This is done by proving an exponential estimate for the Schwartz kernel of the extension operator, and showing that this Schwartz kernel admits a full asymptotic expansion in powers of the line bundle for close parameters. Along the way we prove similar results for the projection onto the holomorphic sections orthogonal to those which vanish along the submanifold. Motivated by possible applications to the invariance of plurigenera problem, our results are proved for manifolds and embeddings of bounded geometry. As an application, we obtain asymptotically optimal $L^{\infty}$-bounds on the holomoprhic extensions from submanifolds, refining previous results of Zhang and Bost., Comment: A version of the extension theorem with respect to the supremum norm is added, see Theorem 1.10. References to earlier works have been added. A minor gap in the proof from Section 4.3 is corrected by a simpler argument