Alexander's Horned Sphere is homeomorphic to a ball. That means that it can be stretched into a ball without puncturing or breaking it (or perhaps it is easier to picture stretching the ball into the horned sphere). The boundary is, therefore, homeomorphic to a sphere. On the other hand, the outer compliment of the solid (what remains of three dimensional space when you take away the solid horned sphere) is not simply connected.
The Jordan Curve Theorem states that if you have a closed curve without self-intersections in the plane, then the plane with the curve may be deformed homeomorphically to a plane with a circle around the origin. If you happen to know what these words mean, then this seems obvious. Alexander's Horned Sphere is, among other things, an example of why the Jordan Curve Theorem cannot be expanded to higher dimensions. If it were possible, you would be deforming something that is not simply connected (the compliment of the horned sphere) into something that is simply connected (the compliment of the ball).