What is the physical meaning of 0 heat capacity? Does it mean that no matter how much heat I put in, I cannot raise the temperature? If so, does this means that if a system is at absolute zero, it cannot be raised in temperature?
Zero heat capacity means an infinitesimally small amount of energy will increase the temperature by an infinitely large amount.
An object has a non-zero heat capacity because it has internal degrees of freedom that it can channel absorbed energy into. The more internal degrees of freedom the higher the heat capacity.
Zero heat capacity emerges as the zero temperature limit of an Einstein solid. We get this limit because the energy spacing of quantum states has to be less than $kT$ for those states to be excited and participate in determining the specific heat. As $T \rightarrow 0$ the spacing is greater than $kT$, and the end result is the solid has no accessible degrees of freedom and therefore a zero heat capacity.
However the moment you add the smallest amount of heat the temperature increases, some states become accessible and the specific heat becomes non-zero. The zero specific heat is more of an idealised limit than a real phenomenon.