The secret behind the operation of the Schmitt Trigger is the voltage on the feedback line. This line is shown in the following animation:

The voltage changes on the feedback line due to the current taken by the driver transistor.
This voltage is reflected back to the first transistor and turns it ON MORE (or turns it OFF). During this time we are not changing the voltage on the base of the first transistor. But the transistor is turning ON MORE!
This creates the “fast action” or “snap action.”