Understanding the physical meaning of interatomic potential and the meaning of its sign

I have been learning about lennard jones potential as a model of interatomic bonds:

$$V(r) = 4ε\left[\left(\frac{σ}{r}\right)^q - \left(\frac{σ}{r}\right)^p\right]$$

But, I've been struggling to understand the physical significance of $$V(r)$$. I understand that it is a form of energy stored in the bond. What I don't understand is what is meant by its sign: what does it mean to have negative energy? I have also heard $$V(r)$$ described as the energy required to fuse the atoms together, and as the energy required to bring them an infinite distance apart, but I don't see how these definitions are reconcilable.

As you have already mentioned, $$V(r)$$ describes the energy within the system due to the interaction of those two atoms. Negative values of $$V$$ could be interpreted as a reduction of the overall energy in the system due to the atom-atom interaction.
Now, let's say the potential has its minimum at position $$r_0$$, so the two atoms would prefer taking this distance while in a bond. If you move them infinitely far apart, their potential energy vanishes as $$V(r) \rightarrow 0$$ as $$r \rightarrow 0$$. Thus the energy it would take to do this would be $$E = \left| V(r_0) - V(r = \infty) \right| = V(r_0)$$