I’m looking into using a CD4021B shift register with an Arduino Uno, and found a schematic online detailing how to use it. However, I am confused as to what the point of the 10kohm resistors is, would the current not just pass straight through from the 5V to the shift register when the switches are closed? To me it seems like the resistors are pointless, however I am rather new to this all so if someone could enlighten me as to their purpose it would be much appreciated.
These are called "pull-down resistors". They prevent the inputs to the IC from floating, and thus being in an indeterminate state. The point is to put the inputs in a known state (logic 0) when the switches are open. When the switches are closed the resistors are effectively "overridden" and a logic 1 is applied to the inputs.
If you have an IC with active-low inputs, then you would want pull-up resistors, which pull the inputs up to Vcc until the switch connects the pin to ground.
The CD4021 is a CMOS device. These have very high input impedance and if left floating (disconnected) could assume either a high or low input condition.
Addition of a pull-down resistor ensures that when the associated switch is opened that the input is pulled to zero rather than leave it in the undefined state.
Closing the switch will, obviously pull the input to +5 V which will be read as logic '1'.
It's a set of drop down resistors to pull the input low when the button isn't pressed.
You can think of the switch and resistor as forming a voltage divider where the switch is either \$0 \Omega\$ or \$\infty\Omega\$ depending on whether it's pressed. so the voltage at the input is either 0V when not pressed or 5V when pressed.