Whenever I hear a plane throttling down quickly, what is happening visually?
Ok I listened and what you are hearing is a turbine powered ag plane.  Most of the rising and decreasing pitch you are hearing that sounds so dramatic is doppler effect from the propeller.  The propeller speed will be constant all this time as the pilot will set the RPM for his activity and leave it, just changing power with throttle.
So in the cockpit, the noise from the prop will more constant without the rising and falling sound, just a steady OOOMMMM of somewhere around 100 hz (three blades at 2000 rpm say), not as intense as the sound heard from the ground, with a roar/hiss from the air passing that rises and falls somewhat with speed as he dives and zooms.
The intensity of the propeller sound will also increase or decrease a lot with the power, but not the overall pitch; because the propeller is constant speed, that remains constant unless the speed is changed, and that won't happen during this maneuvering.
Then the very high pitch zeeeeeeeeee whining sound under the big booming sound is the turbine engine itself. It's sound will change a little bit from within the cockpit as power is added and removed, with some change in the pitch of the whining sound and its intensity.
This video is from the inside of a plane like the one you heard, but the inside sound is being recorded through an active noise cancelling headset, so 80% of the cockpit sound is being suppressed.  But you can hear faintly the airstream, and a bit of the prop's sound and can tell that it's pretty constant to the pilot the entire time.  Way less dramatic for sure.