We consider the Glauber-Kawasaki dynamics on a $d$-dimensional periodic lattice of size $N$, that is, a stochastic time evolution of particles performing random walks with interaction subject to the exclusion rule (Kawasaki part), in general, of non-gradient type, together with the effect of the creation and annihilation of particles (Glauber part) whose rates are set to favor two levels of particle density, called sparse and dense. We then study the limit of our dynamics under the hydrodynamic space-time scaling, that is, $1/N$ in space and a diffusive scaling $N^2$ for the Kawasaki part and another scaling $K=K(N)$, which diverges slower, for the Glauber part in time. In the limit as $N\to\infty$, we show that the particles autonomously make phase separation into sparse or dense phases at the microscopic level, and an interface separating two regions is formed at the macroscopic level and evolves under an anisotropic curvature flow. In the present article, we show that the particle density at the macroscopic level is well approximated by a solution of a reaction-diffusion equation with a nonlinear diffusion term of divergence form and a large reaction term. Furthermore, by applying the results of Funaki, Gu and Wang [arXiv:2404.12234] for the convergence rate of the diffusion matrix approximated by local functions, we obtain a quantitative hydrodynamic limit as well as the upper bound for the allowed diverging speed of $K=K(N)$. The above result for the derivation of the interface motion is proved by combining our result with that in a companion paper by Funaki and Park [arXiv:2403.01732], in which we analyzed the asymptotic behavior of the solution of the reaction-diffusion equation obtained in the present article and derived an anisotropic curvature flow in the situation where the macroscopic reaction term determined from the Glauber part is bistable and balanced.