Push power assisted trolley with one finger
The answer they gave was probably misleading.  

One method, and it is very likely that the one used is generally similar, is to have a "position control" lever that is spring loaded to centre. The lever is vertical and spring loaded to the vertical centre position.  If you push on the lever from behind it moves forwards and the system is designed to move the cart forwards until the lever is again vertical. If the pushing "finger" is stationary the cart will move away until the prior condition again exists. If the finger (and the attached person) moves forwards at a steady space the cart can be arranged to move at that pace. 

The biggest challenge is probably to get a control loop such that the cart neither "shoots away" when the lever is slightly pressed or stops topo suddenly (or moves backwards once pressure is taken off the lever. The user's brain is probably part of thge control loop - getting the cart to move at desired speed up a slope or down a slope probably requires experience in just how hard and fast to push.

Arrangements like this have been "common enough" for a very long time. Rather than a lever they may use a control plate or a "cord" that the user pulls on and the cart follows. These may operate with a caternerary loop (cord hanging in a downwards loop) so the user exerts almost no force at all.