What do the 47k resistors and 10k resistors do in this circuit?
The diodes perform an OR function. 

The middle 10K is there to charge the capacitor back to 3.3V. If you leave it out the charge has to come from the 47K pullups and the current has to come from the reverse current of the diodes. That would not be a great idea. 
I forgot the 47K resistors: as mentioned by Oskar: without you short the power rail. They also make sure the outputs at the RHS are high if no button is pushed. 

You mentioned you want to do the same thing. If it is just multiple reset buttons you can do it with several buttons in parallel. in the circuit above the RHS signals are used.