First, the apple appears to break into 3 parts, not two. Second, the break is obviously not as "clean as a knife" if you slow the video down and watch it carefully. However, the break is much cleaner than if you tried to tear the apple into chunks with your bare hands (unless you happen to be Edward Scissorhands, of course). Apples in general are not designed for high tension, as there is no biological motivation for such a property. On the contrary, apples are basically designed to be consumed by animals, to help propagate the seeds. That is why they have relatively soft flesh on the outside, unlike most nuts.
The other answers have explained why the apple falls apart, so I will try to give an idea for why it breaks into three pieces the way it does. If you look carefully at an intact apple, you will notice that it has a rough radial symmetry about the axis going through the stem. If you study the bottom, in particular, you will notice that quite often, there is a 5-fold symmetry. On top of that, the average apple contains 5-8 seeds. Perhaps you are noticing a pattern here. Biology tends to be very economical (because that is rewarded, and extravagance is penalized with extinction), so many "body plans" look like: "make this widget, and duplicate it N times". For an apple, that appears to be something like: "make this ear-shaped slice and put a seed near the middle, and do that five times in a circle". Of course, it doesn't make the apple fully formed in that shape, but rather starts out growing around the seeds themselves.
When the apple finally breaks, we are then left with the question: "Why does it break where it does?" And as with all mechanical systems, the answer is generally: "That's where the weakest points were." So where are the "weak points" on an apple? Going back to the bottom of the apple, we infer the 5-fold symmetry because of the "bumps" on the bottom, which can range from quite prominent to barely noticeable, if at all. You can imagine that the larger mass/internal surface area near the "peak" of one such bump has higher tensile strength (ability to resist being pulled apart), and that there is a correspondingly lower tensile strength in the "valleys" between the bumps. I would predict that the valleys are thus where the apple is most likely to break. And this also explains why it breaks into 3 pieces, and why the pieces are not the same size. Breaking roughly in half would be the simplest way to release tension force in the apple, but the symmetry makes a clean half-break less likely. Instead, it appears that the apple breaks into more like 2:2:1 fragments. It's not precise, because there isn't a clean division between the "lobes" of the apple.
Given these observations, I would expect that were this experiment to be replicated on an orange, that the orange might be more likely to split cleanly in half, given that oranges typically have 10 sections. In addition, the lobes of an orange are quite distinct, resulting in fairly inhomogeneous internal forces upon rotation. On the other hand, I would not be surprised if large-scale (but subtle to the eye) variations in the shape of the orange caused it to split asymmetrically like the apple, but still along the 10-fold symmetry lines.