greater bond energy, more stable?
Look at it in this way. Consider the energy to break a bond to be arbitrarily assumed to be at $\pu{0 J/mol}$. Now, there are two bonds, $\ce{A-B}$ and $\ce{C-D}$. Let's say they have energies of $\pu{-242 kJ/mol}$ and $\pu{-378 kJ/mol}$. Clearly, the $\ce{C-D}$ bond has the lower energy.

Now, you can say that a stronger bond needs more energy to break it. So, this means that you need to supply $\pu{242 kJ/mol}$ of energy to break $\ce{A-B}$ and $\pu{378 kJ/mol}$ of energy to break $\ce{C-D}$. You can definitely see that $\ce{C-D}$ needs more energy to break it, and to bring the energy to $\pu{ 0 J/mol}$.

Hope this answer helps you.