  The many-body localised phase of quantum systems is an unusual dynamical
phase wherein the system fails to thermalise and yet, entanglement grows
unboundedly albeit very slowly in time. We present a microscopic theory of this
ultraslow growth of entanglement in terms of dynamical eigenstate correlations
of strongly disordered, interacting quantum systems in the many-body localised
regime. These correlations involve sets of four or more eigenstates and hence,
go beyond correlations involving pairs of eigenstates which are usually studied
in the context of eigenstate thermalisation or lack thereof. We consider the
minimal case, namely the second R\'enyi entropy of entanglement, of an initial
product state as well as that of the time-evolution operator, wherein the
correlations involve quartets of four eigenstates. We identify that the
dynamics of the entanglement entropy is dominated by the spectral correlations
within certain special quartets of eigenstates. We uncover the spatial
structure of these special quartets and the ensuing statistics of the spectral
correlations amongst the eigenstates therein, which reveals a hierarchy of
timescales or equivalently, energyscales. We show that the hierarchy of these
timescales along with their non-trivial distributions conspire to produce the
logarithmic in time growth of entanglement, characteristic of the many-body
localised regime. The underlying spatial structures in the set of special
quartets also provides a microscopic understanding of the spacetime picture of
the entanglement growth. The theory therefore provides a much richer
perspective on entanglement growth in strongly disordered systems compared to
the commonly employed phenomenological approach based on the $\ell$-bit
picture.
