That ain't what Text::Balanced does.

Text::Balanced::extract_bracketed( $str, '[]' );
[download]

... will extract the first (and only the first) [...] delimited string, allowing balanced pairs of square brackets within it. It doesn't attempt to then do any parsing within that string. It will also return the remainder of the input string, so you can parse that remainder to, for example, find the next [...] delimited string.

If you need deep parsing, then you need to write a recursive function. Something like this:

use strict;
use warnings;
use Text::Balanced qw(extract_quotelike);
use Data::Dumper;

sub parse_str
{
    my $str = shift;
    $str =~ s/\A\s*//;
    
    # list-like
    if ($str =~ /\A\s*\[/)
    {
        # eat initial [
        $str =~ s/\A\s*\[\s*//;
        
        my @arr;
        while ($str !~ /\A\s*\]/)
        {
            my ($term, $remainder) = parse_str($str);
            push @arr, $term;
            if ($remainder =~ /\A\s*[,\]]/)
            {
                $str = $remainder;
                $str =~ s/\A\s*,\s*//;
            }
            else
            {
                die "Expected comma; got <<<$remainder>>>";
            }
        }
        
        # eat trailing ]
        $str =~ s/\A\s*\]\s*//;
        
        return \@arr, $str;
    }
    
    # quote-like
    elsif ($str =~ /\A['"]/)
    {
        my ($term, $remainder) = extract_quotelike($str);
        $term =~ s/\A['"]//;
        $term =~ s/['"]\z//;
        return ($term, $remainder);
    }
    
    # number-like or bareword-like
    elsif ($str =~ /\A([\w\.]+)/)
    {
        my $term = $1;
        return ($term, substr($str, length($term)));
    }
    
    die "Expected something else; got <<<$str>>>";
}

print Dumper parse_str <<DATA;
[
    'listliteral',
    [
        'ListLiteral',
        [
            '../classSPL_1_1Operator_1_1Instance_1_1ExpressionTree_1_1
+ListLiteral.html',
            1,
            'SPL::Operator::Instance::ExpressionTree'
        ]
    ]
]
DATA
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Good, thank you.

How do I do that?

Recognize that what you have is JSON and use a JSON module :)

#!/usr/bin/perl --
use strict;
use warnings;
use Data::Dump qw/ dd pp /;
use JSON qw/ to_json from_json /;
my $str = q{["listliteral",["ListLiteral",["../classSPL_1_1Operator_1_
+1Instance_1_1ExpressionTree_1_1ListLiteral.html",1,"SPL::Operator::In
+stance::ExpressionTree"]]]};
dd( from_json( $str ) );
__END__
[
  "listliteral",
  [
    "ListLiteral",
    [
      "../classSPL_1_1Operator_1_1Instance_1_1ExpressionTree_1_1ListLi
+teral.html",
      1,
      "SPL::Operator::Instance::ExpressionTree",
    ],
  ],
]
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Oh, but you say I switched the single quotes for the double quotes ... :P

Well, what you do have is also perl, so use Safe

#!/usr/bin/perl --
use strict;
use warnings;
use Data::Dump qw/ dd pp /;
use Safe;
my $str = q{["listliteral",["ListLiteral",["../classSPL_1_1Operator_1_
+1Instance_1_1ExpressionTree_1_1ListLiteral.html",1,"SPL::Operator::In
+stance::ExpressionTree"]]]};
dd( Safe->new->reval( $str ) );
__END__
[
  "listliteral",
  [
    "ListLiteral",
    [
      "../classSPL_1_1Operator_1_1Instance_1_1ExpressionTree_1_1ListLi
+teral.html",
      1,
      "SPL::Operator::Instance::ExpressionTree",
    ],
  ],
]
[download]

very nice, thank you!