The Jordan curve theorem tells us that a continuous loop in the plane with no self-intersection points divides the plane into two connected components, the interior, which is bounded, and exterior, which is unbounded, and both components have that curve as a boundary.
Since both components are open because they are connected they are also path-connected.
But still, that does not seem enough to me.
Because interior cannot have holes (or it can if my intuition is wrong) we should be able to say that interior is at least simply connected.
This is probably proven relatively long time ago, or it can be proven (if true) with some combination of known results from analysis and topology, but since my knowledge is poor in those areas I am not sure what results to combine effectively so as to arrive at this conclusion.
So, how to prove that? Or, where to find a proof?
I suggest you look at the Jordan-Schoenflies theorem, which states that the two connected components of your $\mathbb R^2 \ \backslash \ S^1$ are homeomorphic to the interior and exterior of a unit circle. This implies that the interior component is simply connected.