It is hard to explain what is shown without writing a book on aiki work and anatomy. Like most other things on this site, it must be felt. My words, therefore, will sadly lack a certain precision. If this explanation is interesting to you, but feels incomplete, track down an aikijujutsu school.
The first thing to realize is that what we are looking at is
aiki work. In order to aspire to understand this, one must be aware of the interactions between the elements of the body's anatomy, such as tendons and fascia, in addition to the more obvious bones and muscles. Here is a diagram showing layers in the human body. This is your new world.
The second thing to understand is that all the movements come from the core of the body and are applied in every direction. Most martial artists eventually realize the futility of getting very strong arms if they can just apply the power from the much stronger core muscles, but almost all training remains unidirectional (just apply energy in that direction, then step in that direction). This is different, and is usually referred to as expressing in the six directions.
The first technique we see is called
aiki age (rising energy). The second technique, the "throw" is mostly inconsequential to this conversation, although it also requires a large amount of training.
Tori has to press forward and apply a subtle and definite
kuzushi, which is just a Japanese term to mean unbalance. A direct consequence is that
uke is not grounded and therefore is unable to resist effectively when forces are applied (e.g. try applying a punch to someone when they stand on the tip of their toes).
A requirement of the proper application of
aiki is to seal the opponent's breath by preventing the muscles around the lungs from properly expanding and contracting. The sudden application of this can be heard in the surprised and pained screams. This is different from being hit in the solar plexus because it's not a shock trauma, more like a very sudden squeeze. It's the difference between punching a towel and twisting it. I do not know exactly what causes this to happen inside
The art of
aiki comes into understanding which directions and vectors to use for maximum efficiency. In this video, for instance, after
kuzushi is applied, vectors are applied into
uke's body, turning and locking the scapulas while arching them backwards, preventing
uke from recovering in any way.
Uke is now locked in a position of unbalance.
The second technique is the art of directing
uke from their locked position. It is a matter of sending the right combination of vectors in the right chronology through both hands. A similar technique is shown here between 17:02 and 17:14, where the technique is demonstrated, then shown without an opponent. The difference is that here it is applied through the shoulders, not the hands.
The vector must be applied with the entire body acting in concert. If it is not done so, it will not have the desired effect. Pushing or pulling, for instance, will cause a localized change in
uke's body, and we are looking for a global change. This is why the entire body must act in concert.
Here is a video showing a simple demonstration / explanation (in Japanese) of aiki age which relies on muscle/bone alignment (that is, a lower-level version of what is shown in the original video).
Here is a video (in English) of an aikidoka showing aiki age + aiki sage (rising + descending energy). The explanation is relatively good, though not necessarily as detailed as one who might wish.