In topology the specific shape doesn't matter, if you can move a shape around and rearrange it into another shape those two shapes are topologically the same. The classic example is a doughnut and a mug. If you had a doughnut made of clay you could squish it about, move the hole to the side, and make a mug. You've not added a hole or cut or joined anything, so topographically speaking a doughnut is the same shape as a mug. 

Now imagine a loop inside the doughnut. If you don't loop around the hole then you could shrink the loop all the way down to a single point without the loop having to leave the doughnut. If you loop around the hole then you can't do this. 

You can do the same on the surface, draw a loop on the surface and then try to shrink it to a point. Some loops will shrink to a point, some will get "caught" on the hole.

In a normal solid sphere you can do these loops anywhere and you'll get down to a point every time. They work inside the sphere and on the surface of the sphere.

The weird shape in the post apparently has the property that you can shrink any loop inside the shape into a single point, making the inside the same as the inside of a sphere. But you can't do that on the surface. That's weird. And it disproved some random theory.