Do you know what the term "push-pull" means? If yes, then I won't bore you with the background information.
What is not clear from the schematic (and what might be the core of your question) is that the circuit assumes the input is a 50% duty-cycle squarewave, coming from somewhere else.
Note that only five of the six inverters are used. If you change the part from an 04 to a 14, the sixth inverter can be used as the necessary oscillator.
Please ask a more detailed question about what you don't understand. Also, is this a homework/school question? There is a separate forum for those.
A piezo is simply a crystal. Either man made or natural. When exposed to an electric field it will warp in one direction. When that field is reversed it will warp in the other direction. Push-pull means positive and negative are applied one way to the piezo, then the whole thing reverses. This causes the piezo to make a noise.
The piezo has a natural harmonic resonance that when you hit that frequency the piezo will produce the loudest tone possible.
If you took a piezo and connected (lets say) the positive to a frequency source and the negative to ground then the piezo will charge up and can tend to hold that charge. Even if the frequency driving it drops to or below zero, the piezo will not want to change its state. It will function very poorly as a speaker. That's why the whole thing has to reverse direction, so you can take advantage of the charge in the crystal.
Really, the circuit shown is a bridge configuration, and not a Push-Pull arrangement. The benefits are similar but the circuits are different. What you have here is that when the input signal is high the top side of the piezo is positive and the lower side is held at zero, while when the input signal is low the top side is held low and the bottom side is positive. This doubles the effective voltage swing across the device and results in four times the power delivered to it.