  The gas-phase metallicity of low-mass galaxies increases with increasing
stellar mass ($M_\ast$) and is nearly constant for high-mass galaxies. Theory
suggests that this tight mass-metallicity relationship is shaped by galactic
outflows removing metal-enriched gas from galaxies. Here, we observationally
model the outflow metallicities of the warm outflowing phase from a sample of
seven local star-forming galaxies with stellar masses between
10$^{7}$-10$^{11}$ M$_\odot$. We estimate the outflow metallicities using four
weak rest-frame ultraviolet absorption lines, the observed stellar continua,
and photoionization models. The outflow metallicity is flat with $M_\ast$, with
a median metallicity of $1.0\pm0.6$ Z$_\odot$. The observed outflows are
metal-enriched: low and high-mass galaxies have outflow metallicities 10-50 and
2.6 times larger than their ISM metallicities, respectively. The observed
outflows are mainly composed of entrained ISM gas with at most 22% of the
metals directly coming from recent supernovae enrichment. The metal outflow
rate shallowly increases with $M_\ast$, as $M_\ast^{0.2 \pm 0.1}$, because the
mass outflow rate shallow increases with $M_\ast$. Finally, we normalize the
metal outflow rate by the rate at which star formation retains metals to
calculate the metal-loading factor. The metal-loading factor inversely scales
with $M_\ast$. The normalization and scaling of the metal-loading factor agree
with analytic expressions that reproduce observed mass-metallicity relations.
Galactic outflows fundamentally shape the observed mass-metallicity
relationship.
