Here's the same code with one debug print added to show what's going on at the beginning of each time around the "while 1" loop.

#!/usr/bin/perl

# http://perlmonks.org/?node_id=1189605

use strict;
use warnings;
$| = 1;

sub err { die "ERROR: ", s/\G/<@_>/r, "\n" }

sub crossmap
  {
  my ($left, $right) = @_;
  [ map { my $prefix = $_ ; map $prefix.$_, @$right } @$left ];
  }

sub expr
  {
  my $answer = [ '' ];
  
  print("\ndebug ", s/\G/<<you are here with [@$answer]>>/r, " end  " 
+),
  $answer = /\G\|/gc      ? [ @$answer, @{ expr() } ] :
    /\G\d+/gc     ? crossmap $answer, [ $& ] :
    /\G\[\d+\]/gc ? crossmap $answer, [ $& =~ /\d/g ] :
    /\G\(/gc      ? ( crossmap($answer, expr()),
                    /\G\)/gc || err "missing ')'")[0] :
    return $answer while 1;
  }

while(<DATA>)
  {
  chomp;
  print "$_ => ";
  my $answer = expr();
  /\G\z/gc or err "incomplete parse";
  local $" = ',';
  print "@$answer\n";
  }

__DATA__
(65|70)
(3[678]|4[1678])
5[45]
(6[4569]|7[01])
[download]

It prints the following ( showing both where the current position in the input is (the \G) and the contents of the variable $answer at the time).:

(65|70) => 
debug <<you are here with []>>(65|70) end  
debug (<<you are here with []>>65|70) end  
debug (65<<you are here with [65]>>|70) end  
debug (65|<<you are here with []>>70) end  
debug (65|70<<you are here with [70]>>) end  
debug (65|70<<you are here with [65 70]>>) end  
debug (65|70)<<you are here with [65 70]>> end  65,70
(3[678]|4[1678]) => 
debug <<you are here with []>>(3[678]|4[1678]) end  
debug (<<you are here with []>>3[678]|4[1678]) end  
debug (3<<you are here with [3]>>[678]|4[1678]) end  
debug (3[678]<<you are here with [36 37 38]>>|4[1678]) end  
debug (3[678]|<<you are here with []>>4[1678]) end  
debug (3[678]|4<<you are here with [4]>>[1678]) end  
debug (3[678]|4[1678]<<you are here with [41 46 47 48]>>) end  
debug (3[678]|4[1678]<<you are here with [36 37 38 41 46 47 48]>>) end
+  
debug (3[678]|4[1678])<<you are here with [36 37 38 41 46 47 48]>> end
+  36,37,38,41,46,47,48
5[45] => 
debug <<you are here with []>>5[45] end  
debug 5<<you are here with [5]>>[45] end  
debug 5[45]<<you are here with [54 55]>> end  54,55
(6[4569]|7[01]) => 
debug <<you are here with []>>(6[4569]|7[01]) end  
debug (<<you are here with []>>6[4569]|7[01]) end  
debug (6<<you are here with [6]>>[4569]|7[01]) end  
debug (6[4569]<<you are here with [64 65 66 69]>>|7[01]) end  
debug (6[4569]|<<you are here with []>>7[01]) end  
debug (6[4569]|7<<you are here with [7]>>[01]) end  
debug (6[4569]|7[01]<<you are here with [70 71]>>) end  
debug (6[4569]|7[01]<<you are here with [64 65 66 69 70 71]>>) end  
debug (6[4569]|7[01])<<you are here with [64 65 66 69 70 71]>> end  64
+,65,66,69,70,71
[download]

I hope this is helpful. At each debug output you can see what the /\G.../gc has stepped over and what the new value of $answer is.

My problem with explaining this type of parser is that I have been working with parsers like this for well over a year, and it all comes as second nature to me. If you have more specific questions I'll be willing to take a shot at answering them.