The kickdown cable is used to regulate a pressure in transmission valve body. This is called throttle pressure.

The throttle pressure is effectively a pressure that "tells" shift valves in transmission how hard you pushing throttle and these shift valves now have a contest to see if governor pressure or throttle pressure is going to win. This pressure is also used to apply the clutches/brakes that engage a gear and the higher pressure goes along with higher engine power at higher throttle. Firmer shifts are a result of higher throttle pressure.

If throttle pressure wins the contest the trans remains in lower gear, if governor pressure wins the trans upshifts.

Governor pressure is directly related to driveshaft, and thus road speed.

If you tighten cable you increase throttle pressure and the whole shift point/road speed "map" goes higher.

If you loosen cable the shift point map moves lower.

The trans throttle cable (kickdown cable) also depresses a valve if you (or the throttle spool) pull the cable all the way out past that hard spot which is a detent to make you aware of the actual kickdown feature.

The kickdown valve increases the throttle pressure drastically above the linear rate that you get from the rest of the throttle pedal travel range and makes the gearbox goes to lowest possible gear allowed at the road speed you are at when you activate it.