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That is cause a almost all the mass of the atom (the protons and neutrons) are bound together in a very very very tiny space.. why? cause they are held by this amazingly strong force which we call as the Strong nuclear force.. The most strongest force in a nature (as per the current laws of physics :D )
Actually we probably don't know. The strong force is attractive and produces a negative energy when engaged reducing the total mass of the interacting particles (E=mc^2). For example the proton is made up of three quarks each much more mass than a proton but interacting via the strong force end up being considerably less massive.
Actually the strong force could be used as a rationale for high density but doesn't explain the mass which does contribute to density.