TTL inverter 74LS14 has an internal 16-Kohm, pull-up resistor that pulls the gate input high when the switch is open. As you close the switch, the 4.7uF capacitor discharges on the first contact. If the switch contacts bounce open, the internal resistor limits the capacitor`s recharge to a rate sufficiently slow to prevent an undesired gate transition before the contacts again close.
Note that the circuit correctly debounces the switch for both opening and closing. If you add an external pull-up resistor, you can use a CMOS Schmitt-trigger gate, 74HC14, and a smaller, 0.1uF, capacitor.