You are modifying @arr inside the loop in such a way that $thing containing 'a' at each iteration. It's easier to see with some modifications:

use warnings;
use strict;
use diagnostics;

my @arr = ('a'..'z');
print "@arr\n";
foreach my $thing (@arr) {
    print "[$thing]";
    my $ele = pop @arr;
    my $stuff = unshift @arr, $ele;
    print "$ele - @arr\n";
}
__END__
a b c d e f g h i j k l m n o p q r s t u v w x y z
[a]z - z a b c d e f g h i j k l m n o p q r s t u v w x y
[a]y - y z a b c d e f g h i j k l m n o p q r s t u v w x
[a]x - x y z a b c d e f g h i j k l m n o p q r s t u v w
[a]w - w x y z a b c d e f g h i j k l m n o p q r s t u v
[a]v - v w x y z a b c d e f g h i j k l m n o p q r s t u
[a]u - u v w x y z a b c d e f g h i j k l m n o p q r s t
[a]t - t u v w x y z a b c d e f g h i j k l m n o p q r s
[a]s - s t u v w x y z a b c d e f g h i j k l m n o p q r
[a]r - r s t u v w x y z a b c d e f g h i j k l m n o p q
[a]q - q r s t u v w x y z a b c d e f g h i j k l m n o p
[a]p - p q r s t u v w x y z a b c d e f g h i j k l m n o
[a]o - o p q r s t u v w x y z a b c d e f g h i j k l m n
[a]n - n o p q r s t u v w x y z a b c d e f g h i j k l m
[a]m - m n o p q r s t u v w x y z a b c d e f g h i j k l
[a]l - l m n o p q r s t u v w x y z a b c d e f g h i j k
[a]k - k l m n o p q r s t u v w x y z a b c d e f g h i j
[a]j - j k l m n o p q r s t u v w x y z a b c d e f g h i
[a]i - i j k l m n o p q r s t u v w x y z a b c d e f g h
[a]h - h i j k l m n o p q r s t u v w x y z a b c d e f g
[a]g - g h i j k l m n o p q r s t u v w x y z a b c d e f
[a]f - f g h i j k l m n o p q r s t u v w x y z a b c d e
[a]e - e f g h i j k l m n o p q r s t u v w x y z a b c d
[a]d - d e f g h i j k l m n o p q r s t u v w x y z a b c
[a]c - c d e f g h i j k l m n o p q r s t u v w x y z a b
[a]b - b c d e f g h i j k l m n o p q r s t u v w x y z a
[a]a - a b c d e f g h i j k l m n o p q r s t u v w x y z
[download]

Before the loop @arr contains 'a'..'z' as expected.

First iteration: $thing ($arr[0]) contains 'a', $ele ($arr[25]) contains 'z' which goes to the beginning and now 'a' is at $arr[1].

Second iteration: $thing ($arr[1]) contains 'a', $ele ($arr[25]) contains 'y' which goes to the beginning and now 'a' is at $arr[2].

Wash, rinse, repeat until you've reached $arr[25].

update: removed meaningless comment