Why should I call detachInterrupt at the start of an interrupt service routine? 

To understand the explanation below, please consider this example sketch. This sketch is used to debounce a rotary encoder:

(Original is sourced from the Arduino Step by Step Your Complete Guide Github repository)

//Original sketch: https://bigdanzblog.wordpress.com/2014/08/16/using-a-ky040-rotary-encoder-with-arduino/
//Modified by Peter Dalmaris, July 2015
const int PinCLK=2;                   // Used for generating interrupts using CLK signal
const int PinDT=3;                    // Used for reading DT signal
const int PinSW=8;                    // Used for the push button switch
volatile long     virtualPosition  =0;  // must be volatile to work with the isr
void isr0 ()  {
  detachInterrupt(0);
  if (!digitalRead(PinDT))
      virtualPosition++; 
  else
      virtualPosition--;
  attachInterrupt (0,isr0,RISING);
  } // ISR0
void setup ()  {
  pinMode(PinCLK,INPUT);
  pinMode(PinDT,INPUT);  
  pinMode(PinSW,INPUT);
  attachInterrupt (0,isr0,RISING);   // interrupt 0 is always connected to pin 2 on Arduino UNO
  Serial.begin (9600);
  Serial.println("Start");
}
void loop ()  {
  int lastCount = 0;
  while (true) {
      if (!(digitalRead(PinSW))) {        // check if pushbutton is pressed
        virtualPosition = 0;            // if YES, then reset counter to ZERO
        while (!digitalRead(PinSW)) {}  // wait til switch is released
        Serial.println("Reset");        
        }
      if (virtualPosition != lastCount) {
        lastCount = virtualPosition;
        Serial.print("Count:");
        Serial.println(virtualPosition);
        }
   } // while
}

When an interrupt “fires”, the Arduino goes into the interrupt service routine (isr).

Imagine that the Arduino is already executing code in the isr, when another interrupt fires. What do you think it is that the Arduino will do next?

The Arduino will stop what its doing and jump back at the start of the ISR.

But that’s not what we want. Normally, the ISR function contains code that is critical to our application, and we want that code to complete its execution always, regardless if another interrupt arrives.

To prevent repeated calls to the ISR before a previous ISR has finished, we must turn off interrupts as soon as one is received. That is what we do in line 12 with the “detachInterrupt” function.

But then, just before the Arduino finished the execution of the ISR, we turn interrupts back on so that the next interrupt is captured. Thats what the call to attachInterrupt does in line 23.