If Space is expanding between the Galaxies then why isn't it also expanding between the Stars within the Galaxies as well ? In fact why isn't Space expanding within our Solar System ?
The Universe is remarkably homogeneous, i.e. the same everywhere, and isotropic, i.e. the same in every direction (NB homogeneity does not imply isotropy and there are toy cosmological models which are homogeneous but not isotropic) and this is the underlying physical assumption of big bang cosmology.
When modelling the Universe the simplification of it being 100% homogeneous and isotropic is most often used as because on the very largest scale (i.e. the scale of the observable Universe) this is very close to being true and any deviations from this are small enough that they do not effect most of the properties being studied. In 100% homogeneous and isotropic Universe, expansion takes place at all places and at all scales.
However we know that the Universe is not 100% homogeneous and isotropic, especially on a smaller scale; for example the centre of a star is a very different place from intergalactic space. It is a fair question then to ask in view of this how does expansion manifest itself on smaller scales.
We know that in gravitationally bound systems such as our local group (which includes Andromeda) matter has the tendency to be drawn closer by gravity rather than to move away as objects do in an expanding Universe, so it seems our local group is not expanding. Still, it is very tempting to see expansion on this scale as being like a small repulsive force which acts against gravity, but is not strong enough to overcome gravity. So in this sense even the local group would be expanding
However the view of expansion acting like a small repulsive force on a small scale is not necessarily the right view. A counterexample would be the Einstein-Strauss Swiss-cheese model where expansion only takes place in deep space and not in the space around stars (modeled as vacuoles). The Einstein-Strauss model though cannot be taken as the last word on the subject as the way it models stars in space is not that realistic.
Overall then how expansion manifests itself on a smaller scale is still very much an open question and we just don't know what tiny corrections if any it would make on the dynamics of a very small system such as the solar system.
Actually, space is expanding everywhere. It is more noticeable between galaxies since space between galaxies is much larger than space between stars and space between the planets in the solar system. A 1% expansion of space only represents 1.5 million kilometers between the Sun and the Earth. But the same 1% expansion between two galaxies that are four billion light-years away means that they are now 40 light-years farther away. And 40 light-years is ten times the distance from the Sun to the nearest star Proxima Centaury.