  The cluster mass function traces the growth of linear density perturbations
and provides valuable insights into the growth of structures, the nature of
dark matter, and the cosmological parameters governing the Universe. The
primary science goal of eROSITA, on board the {\it Spectrum Roentgen Gamma
(SRG)} mission, launched in 2019, is to constrain cosmology through the
evolution of cluster mass function. In this paper, we present the cosmological
constraints obtained from 5259 clusters of galaxies detected over an area of
12791~deg$^2$ in the Western Galactic Hemisphere of the eROSITA's first All-Sky
Survey (eRASS1). The common footprint region between the eROSITA Survey and
DES, KiDS, and HSC surveys is used for calibration of the scaling between X-ray
count rate and their total mass through measurements of their weak
gravitational lensing signal. eRASS1 cluster abundances constrain the
$\Lambda$CDM parameters, which are the energy density of the total matter to
$\Omega_{\mathrm{m}}=0.29^{+0.01}_{-0.02}$, and the normalization of the
density fluctuations to $\sigma_8=0.88\pm0.02$ and their combination yields
$S_8=\sigma_8 (\Omega_\mathrm{m} / 0.3)^{0.5}=0.86\pm0.01$, consistent and at a
similar precision with the state-of-the-art CMB measurements. eRASS1
cosmological experiment places a most stringent upper limit on the summed
masses of left-handed light neutrinos to $\sum m_\nu< 0.22\mathrm{~eV}$ (95\%
confidence interval). Combining eRASS1 cluster abundance measurements with CMB
and ground-based neutrino oscillation experiments, we measure the summed
neutrino masses to be $\sum m_\nu=0.08_{-0.02}^{+0.03}\mathrm{~eV}$ or $\sum
m_\nu=0.12_{-0.01}^{+0.03}\mathrm{~eV}$ depending on the mass hierarchy
scenario for neutrino eigenstates. eRASS1 cluster abundances significantly
improve the constraints on the dark energy equation of state parameter to
$w=-1.12\pm0.12$. (ABRIDGED)
