My question is, why is the 10k resistor needed after the switch? Why can't it just go straight to ground? The switch's output just connects to a digital pin, it doesn't care how much voltage there is, correct?
The resistor is needed so when the switch is closed you don't short the supply
to ground. Switch open the resistor pulls the signal to ground and no current
flows, when the switch is closed it forms a resistive divider with the 10k
(switch contacts might have 0.01 to 0.1 ohms resistance or so), thus pulling
the signal to Vcc, but with only 0.5mA flowing through the 10k resistor.
Replace the 10k with a wire and you'd simply overload the supply completely and
perhaps weld the switch contacts shut.