What causes "ear rumbling"?
I'm able to create rumbling in my ears at will. Unlike the poster of the video, I don't need to yawn to do so; I can do it without my face appearing to move. I actually discovered this alongside another ability of mine when I was little: the ability to create warmth flowing from the base of my neck outwards into the rest of my body. After some research I figured out that the warmth I'm able to create is actually part of voluntary piloerection (its use for warmth is mentioned here), while the ability to create a rumbling sound is simply the contraction of the tensor tympani, a muscle in the ear. Voluntary control over either one of these things is supposed to be rare, but I've found claims of several other people online who can do both (example1, example2).
Anyways, Wikipedia gives a brief explanation of how the ear rumbling works, which also specifically mentions &quot;yawning deeply&quot; as a trigger:

Contracting muscles produce vibration and sound. Slow twitch fibers produce 10 to 30 contractions per second (equivalent to 10 to 30 Hz sound frequency). Fast twitch fibers produce 30 to 70 contractions per second (equivalent to 30 to 70 Hz sound frequency). The vibration can be witnessed and felt by highly tensing one's muscles, as when making a firm fist. The sound can be heard by pressing a highly tensed muscle against the ear, again a firm fist is a good example. The sound is usually described as a rumbling sound.
Some individuals can voluntarily produce this rumbling sound by contracting the tensor tympani muscle of the middle ear. The rumbling sound can also be heard when the neck or jaw muscles are highly tensed as when yawning deeply. This phenomenon has been known since (at least) 1884.

It also seems like this type of noise can happen with conditions such as Tonic Tensor Tympani Syndrome.
The tensor tympani connects directly to the handle of the malleus, so I'd assume that's why it can be heard by the person doing it. Apparently this also causes the eardrum to move. This means it's classified as objective tinnitus.
Here are some relevant studies I found:

Voluntary contraction of the tensor tympani muscle
and its audiometric effects
Voluntary Eardrum Movement : A Marker for Tensor Tympani Contraction? (paywall)
Tonic contractions of the tensor tympani muscle: a key to some
non-specific middle ear symptoms? Hypothesis and data from
temporal bone experiments (paywall)