Is limb darkening of the Sun because at centre we are directly looking the core and different zones, whereas at limbs we are seeing only the photospheric part (and less beneath photospheric part)? Or there is some other physics behind this phenomenon?
The visible surface of the Sun is defined by that location within the Sun's atmosphere from which photons produced in the gas can escape towards the Earth. That location is determined by the amount of gas between it and the Earth, since that gas would reabsorb photons if there were enough of it.
Now you need to think about the geometry of a roughly, spherically symmetric Sun. When we look at the solar centre, photons can escape from a certain layer in its interior. That layer is nowhere near the core of the Sun, it is the solar photosphere at a temperature of approximately 6000K and at a radius of $6.96\times 10^8$ m from the centre.
Now imagine looking towards the limb. The same quantity of intervening gas is reached by the time you get to a layer in the Sun that is at a larger radius than at the solar centre. Because the temperature decreases as you move outwards in the solar photosphere, this layer is cooler and has a lower surface brightness (surface brightness is proportional to $T^4$). As a result, the solar centre looks hotter and brighter than the limb of the Sun.