Why is that any sensor signal need to be hooked up to the interrupt pins on Andruino boards?
The interrupt pins are used to notify the MCU that a device needs attention. The INTx pins are chosen first because they are easy to program for; each gets a dedicated interrupt vector and hence ISR.

There is another type of external interrupt though, the Pin Change interrupt. This can be invoked via any of the PCINTxx pins on the device; on the '328 any pin can act as an interrupt.

The main difference between INT and PCINT is that PCINTs are grouped into banks of 8, which means that once the ISR has been invoked the appropriate pins need to be checked for their level, which furthermore means that it is possible to misread an interrupt if a bank is shared by multiple devices and the interrupt pulse itself is very short.

Further details can be read in the "External Interrupts" and "Interrupts" sections of the MCU datasheet.