What are all the multiple pins on PSU connectors for? How does a 24-pin MB connector work exactly?
As Passerby says, the main reason why there are several wires for any given voltage is that a computer may require tens of amps, so one wire by itself isn't enough.

The reason for all the voltages is largely historical.  For many years, the standard voltage for digital logic circuits was 5V.  Many circuits still run at 5V.

However, later chip designs could be made to work at lower voltages, and would run cooler when powered at a lower voltage.  So 3.3V was added to the power supply.

+12V is mainly used for powering motorized things, like disk drives and fans.  It's also needed for RS232 serial ports.  RS232 also requires -12V, so power supplies often have a low-current -12V line as well.

The 5Vsb line is a "standby" power line, to keep some parts of the computer powered on while the computer is in "sleep" mode.