  The large-scale behaviour of entanglement entropy in finite-density states,
in and out of equilibrium, can be understood using the physical picture of
particle pairs. However, the full theoretical origin of this picture is not
fully established yet. In this work, we clarify this picture by investigating
entanglement entropy using its connection with the large-deviation theory for
thermodynamic and hydrodynamic fluctuations. We apply the universal framework
of Ballistic Fluctuation Theory (BFT), based the Euler hydrodynamics of the
model, to correlation functions of \emph{branch-point twist fields}, the
starting point for computing R\'enyi entanglement entropies within the replica
approach. Focusing on free fermionic systems in order to illustrate the ideas,
we show that both the equilibrium behavior and the dynamics of R\'enyi
entanglement entropies can be fully derived from the BFT. In particular, we
emphasise that long-range correlations develop after quantum quenches, and
accounting for these explain the structure of the entanglement growth. We
further show that this growth is related to fluctuations of charge transport,
generalising to quantum quenches the relation between charge fluctuations and
entanglement observed earlier. The general ideas we introduce suggest that the
large-scale behaviour of entanglement has its origin within hydrodynamic
fluctuations.
