We develop an analytical approach for the study of the quench dynamics of the anisotropic Heisenberg model (XXZ model) on the infinite line. We present the exact time-dependent wave functions after a quench in an integral form for any initial state and for any anisotropy Δ by means of a generalized Yudson contour representation. We calculate the evolution of several observables from two particular initial states: starting from a local Néel state we calculate the time evolution of the antiferromagnetic order parameter-staggered magnetization; starting from a state with consecutive flipped spins (1) we calculate the evolution of the local magnetization and express it in terms of the propagation of magnons and bound state excitations, and (2) we predict the evolution of the induced spin currents. These predictions can be confronted with experiments in ultracold gases in optical lattices. We also show how the "string" solutions of Bethe ansatz equations emerge naturally from the contour approach.