  We study the structural evolution of isolated star-forming galaxies in the
Illustris TNG100-1 hydrodynamical simulation, with a focus on investigating the
growth of the central core density within 2 kpc ($\Sigma_{*,2kpc}$) in relation
to total stellar mass ($M_*$) at z < 0.5. First, we show that several
observational trends in the $\Sigma_{*,2kpc}$-$M_*$ plane are qualitatively
reproduced in IllustrisTNG, including the distributions of AGN, star forming
galaxies, quiescent galaxies, and radial profiles of stellar age, sSFR, and
metallicity. We find that galaxies with dense cores evolve parallel to the
$\Sigma_{*,2kpc}$-$M_*$ relation, while galaxies with diffuse cores evolve
along shallower trajectories. We investigate possible drivers of rapid growth
in $\Sigma_{*,2kpc}$ compared to $M_*$. Both the current sSFR gradient and the
BH accretion rate are indicators of past core growth, but are not predictors of
future core growth. Major mergers (although rare in our sample; $\sim$10%)
cause steeper core growth, except for high mass ($M_*$ >$\sim$ $10^{10}
M_{\odot}$) mergers, which are mostly dry. Disc instabilities, as measured by
the fraction of mass with Toomre Q < 2, are not predictive of rapid core
growth. Instead, rapid core growth results in more stable discs. The cumulative
black hole feedback history sets the maximum rate of core growth, preventing
rapid growth in high-mass galaxies ($M_*$ >$\sim$ $10^{9.5} M_{\odot}$). For
massive galaxies the total specific angular momentum of accreting gas is the
most important predictor of future core growth. Our results suggest that the
angular momentum of accreting gas controls the slope, width and zero-point
evolution of the $\Sigma_{*,2kpc}$-$M_*$ relation.
