firstly understand the context which is smooth manifolds, for simplicity imagine a 2d manifold embedded in 3d space - so a sheet of rubber that can pass through itself but can't kink or do any funny business, just like in that sphere inversion video.

the definition of a manifold is basically that it can be built out of patches (sheets of rubber in our analogy), for instance to make a sphere, we need two sheets of rubber (ignore the actual logistics of the deformation required).

Now say that our sheets of rubber come with a textured and a smooth side, there are two ways to attach the sheets of rubber to make a sphere, one of which produces a sphere which is entirely smooth on the outside. This is what we mean by orientable, we can build it out of patches with a consistent "outside".

Consider the counterexample of a mobius strip, which we construct from a single strip of rubber by attaching one end to the other "backwards" (rough-smooth). Since we have defined it this way, it cannot be orientable. The klein bottle is another example, but somewhat cooler than the mobius strip since its a surface without edges.

There are many other definitions of orientable depending on the context, since manifolds are a lot more general than I have shown you here.

I don't know what orientable manifolds have to do with being responsible.