What is the physical explaination for the presence electrostatic forces (just like we have for gravity)?

As explained by Einstein, the gravitational force is a result of the curvature of spacetime. The electrostatic force is caused by the exchange of photons (although physicists have hypothesised that the gravitational force is mediated through the exchange of gravitons, but this has never been verified). That is how the electric force "travels". And if you vary the position of one particle, the particles do not feel the change instantaneously. Rather, at the speed of light, $c$. Similarities between these two forces are the fact that (at a classical level) the expression for their forces are both inverse-square in design. In the context of your question though, Einstenian Gravity and the electrostatic force are not similar at all.

The electric and magnetic fields are respectively the space-time and purely spacelike curvature of the photon field. Just like the curvature 2-form $Omega$ is the curvature of the spin connection field $omega$ ($Omega=d_omega omega=d omega+omegawedgeomega$) in the tetradic formulation of general relativity. By «space-time» curvature I mean that if you have an electrostatic field acting on a charged particle initialy at rest in the reference frame of the source of our field, it will move in space as time goes.

The electrostatic force in QED can be interpreted as an exchange of virtual photons. The field is massless so the information of any change in our field travels at the speed of light.

If you have two particles hundred kilometers apart, and if one moves toward the other, then from our referential the other does not move. This means that the electric potential created by it is static and then the first particle will instanteanously feel the change in potential, because this change in potential is right at the position of the particle. On the other hand, the other particle will feel a change in the potential from the first only a time after it moved. This is because from our referential, the potential from the first particle is changing in time. And so the information of this change travels at the speed of light. If one changes of referential from the second to the first particle, their roles will be exchanged and so the conclusions.

Finaly to continue Dr jh's comment, in small metric perturbation regime, general relativity can be formulated analogously to Maxwell's equations. And this is called gravitoelectromagnetism.