so the picture, as a simple planar object with Einstein's face printed on it, passes behind a 3-D black hole (or if you like, the black hole passes in front of the stationary object - it's all relative..)

To simplify things computationally the simplest type of black hole is chosen - the Schwartzchild black hole which is spherically symmetric, doesn't rotate or move, and has no electric charge. Schwartzchild derived the metric (or the spacetime) for this, which is basically the funnel shape to a singularity, which we all know and love ..

What the GIF shows is (an approximation of) how the resultant scene - the image of the situation - would look to us at some distance away, where we are well beyond the event horizon and are not going to get sucked in. As David said, at such distances the influence of a black hole is simply like that of a massive star - no need to worry.

If you want more detailed info, check out the web page of one of the pioneers of this work, Werner Benger