$b{$readID} || []
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When there are no entries for $readID in DUMP_B, $b{$readID} becomes undef, appending || [] handles this special case replacing undef by a reference to an empty array so that we can dereference it later as @$ac.

A simpler version of the code follows:

my %b;

open(B, "<" . DUMP_B) || die("Could not open file \n");
while (<B>) {
  chomp;
  my ($id, $ac) = split /\|/, $_;
  push @{$b{$id}}, $ac;
}

open( OUTPUT, '>', 'INACTIVE_LIST' ) || die "Could not write to file\n
+";
open(A, "<" . DUMP_A) || die("Could not open file \n");
while(<A>) {
  chomp;
  s/\s+//g;
  if ($b{$_}) {
    for my $ac (@{$b{$_}}) {
      print OUTPUT "$_|$ac\n"
    }
  }
}
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BTW, when programming in Perl it is very unusual to use C-like (or Java-like) for loops. Instead, for (@array) { ... } and for (0..$top) { ... } are used.