Acid-Base reaction:
CaCO3 (calcium carbonate) + 2 HOOCCH3 (acetic acid) = Ca(OOCCH3)2 (calcium acetate) + H2CO3 (carbonic acid)
The CaCO3 from the egg shell is a hard solid (it is the main component of for instance stalactites) and when it reacts with the acetic acid that vinegar contains forms Ca(OOCCH3)2 that is a powdery solid soluble in H2O. The carbonic acid released is unstable and decomposes into CO2 (gas bubbles around the egg shell) and H2O.
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