  Entanglement and entropy are key concepts standing at the foundations of
quantum and statistical mechanics, respectively. In the last decade the study
of quantum quenches revealed that these two concepts are intricately
intertwined. Although the unitary time evolution ensuing from a pure initial
state maintains the system globally at zero entropy, at long time after the
quench local properties are captured by an appropriate statistical ensemble
with non zero thermodynamic entropy, which can be interpreted as the
entanglement accumulated during the dynamics. Therefore, understanding the
post-quench entanglement evolution unveils how thermodynamics emerges in
isolated quantum systems. An exact computation of the entanglement dynamics has
been provided only for non-interacting systems, and it was believed to be
unfeasible for genuinely interacting models. Conversely, here we show that the
standard quasiparticle picture of the entanglement evolution, complemented with
integrability-based knowledge of the asymptotic state, leads to a complete
analytical understanding of the entanglement dynamics in the space-time scaling
limit. Our framework requires only knowledge about the steady state, and the
velocities of the low-lying excitations around it. We provide a thorough check
of our result focusing on the spin-1/2 Heisenberg XXZ chain, and considering
quenches from several initial states. We compare our results with numerical
simulations using both tDMRG and iTEBD, finding always perfect agreement.
