I find it helps to unpack this sort of code and figure out names for intermediate variables. It also helps to add a little white space.

use strict;
use warnings;

my @input = (qw/a b c d/);

permutation('', @input);

sub permutation {
    my ($perm, @set) = @_;
    
    if (!@set) {
        print "$perm\n";
        return;
    }
        
    for my $partition (0 .. $#set) {
        my $firstEnd = $partition - 1;
        my $lastStart = $partition + 1;
        my @newSet = @set[0 .. $firstEnd];
        
        push @newSet, @set[$lastStart .. $#set];
        permutation($perm . $set[$partition], @newSet);
    }
}
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This is a recursive algorithm (the function calls itself). With recursion there are two important parts - the recursion process and the stop recursing condition. The stop recursing condition is that the @set parameter is empty. In the stop condition processing $perm is printed (the result of the current recursive branch) and the function returns.

For the recursive processing the loop takes each element from @set and concatenates it on to the passed in $perm then passes the new string and @set with the current element removed down to the next recursion level. Because each recursion level removes one element from @set and tacks it onto $perm eventually all elements are removed from @set and tacked onto $perm and each of the ordering of the passed in @set are generated.