The shell of an egg (typically a chicken egg) is made up of primarily calcium carbonate. If you soak this egg shell in vinegar (which is about 4% acetic acid), you start a chemical reaction that dissolves the calcium carbonate shell. The acetic acid reacts with the calcium carbonate in the egg shell and releases carbon dioxide gas that you see as bubbles on the shell.
CaCO3 (s) + 2 HC2H3O2 (aq) → Ca(C2H3O2)2 (aq) + H2O (l) + CO2 (g)
The egg insides remain intact and are held together by the two fragile membranes just inside the shell.