Spatial disorientation, spins, and illusions are complex subjects each. In this case, the initial explanation in the video is straightforward. If one gets into a particular flight attitude, especially slowly, and doesn't feel one's self getting there (slowly creeping into a bank), the body doesn't know it's in that attitude (eg, bank). The body thinks nothing has occurred to move it from level flight. The sensory perception in the inner ear hasn't indicated a change from level to the bank, so when one sees the instrument indicating a bank, the natural inclination is to believe the one instrument that's always been true...the brain and the body.
Of course, this isn't true. The instrument is correct. But one perceives the instrument to be incorrect; after all, how did one get into that bank, without sensing it. If one suddenly rights the airplane according to the instrument, or in other words returns the airplane to level flight, now the body senses a movement. This movement is contrary to what the body believes; the body believes it was level, but the movement now is sensed as a bank.
When the body senses a bank, but the instrument is showing level, this is confirmation bias: the body believed it was correct before (even though it wasn't), and thus that the instrument was incorrect. Now, having moved the airplane back to level flight, the body is receiving a signal of movement, and believes itself now banked...the instrument says it's not in a bank. This confirms the previous believe, or is confirmation bias, implying that the body was correct all along, and the instrument wrong. This furthers distrust of the instrument.
What has really occurred is that the pilot has returned the airplane to level flight, but perceives himself or herself to have entered a bank. Now a sensory conflict exists. Which is correct? Instrument training will tell the pilot to believe his instruments. A lifetime of standing upright will tell the pilot to believe his inner ear. If the pilot does not follow the instrument indications, he can roll back into the bank where he was, and even steepen it; each upset or correction can make the sensation worse.
In the process, the upset is further worsened if there are changes in pitch, or in speed. Changes in speed will usually be perceived as changes in pitch. As bank angle increases, these changes are further complicated by increasing G load, or unload, which may be perceived as pitching moments, further complicating the matter, and a pilot in a bank who begins to apply pitch can further complicate the matter, steepening the bank, unloading the wing, increasing or decreasing senses of bank, speed, and pitch.
The immediate solution is simple in description, harder when experiencing strong conflicting sensations (disorientation). Fly the instruments. Easier said than done, especially initially when one wants to believe one's own senses. I have seen very experienced pilots, on several occasions, become disoriented even in visual conditions, or visual conditions with illusions. In one case, in a 747, I took the controls briefly when the pilot became disoriented with a sloping horizon illusion. We were at night, departing Bahrain, and a combination of dark, fires on oil rigs in the gulf, and stars in the sky combined to disorient the pilot. It only took a moment for him to regain his bearings, but it does happen. I saw the same thing once flying into Liege in the daytime in reduced visibility, but still visual conditions, when turning final. I have experienced the same thing myself, both as a student and as a seasoned professional, and it's very real. The solution: fly the instruments, but again, easier said in some cases, than done, and it doesn't change the strong, sometimes seemingly overpowering sensation that what the body feels is correct, and what the instruments are telling you, is not.
This is the reason we don't just look at one instrument, but crosscheck them, and interpret them: to ensure that what we're seeing is correct, and that we're seeing the whole picture.
In a spin, the effects are magnified significantly, with numerous factors coming into play. To complicate that, spins are not stead-state events, staying the same; they change dynamically due to a number of factors, and physical and psychological factors combine to cause very powerful illusions that can lead one to make incorrect choices and further aggravate the situation, rather than effect a recovery.