What does it actually mean by stability of an atom
First you need to understand what stability is. Normally, octet rule is based on noble gas configuration. Since, noble gases hardly react with other atoms, and they have 8 electrons as their shell configuration, thus we consider them "stable".

So, taking noble gas 8 electrons shell configuration, we tend to generalise all other atoms/elements to have 8 electron filled orbit to gain stability.

You must understand that scientist create models to explain natural phenomena. Rules are  designed to explain already existing phenomena.  Stability truly means "less energy" or inert behaviour where the atom neither gains nor lose electrons to other atoms/molecules. The lesser the energy of the atom/system, more the stability.

Thus your doubt on "does not explain why an atom is stable when it obeys those rules" is very vague. Think of inert gases, their configuration, their inert nature and you will understand.