From the beginning...

$config is a hashref. It can be dereferenced to get the hash itself by prefixing it with %: %$config. Except there'll be precedence issues, so %{$config} is better. The values in the hash can be accessed via their keys like so ${$config}{groups}. But that's too longwinded. The "pointer" operator -> is used to shorten %{$config}{groups} to $config->{groups}.

The value associated with that particular key is another hashref, so that is accessed with another use of -> shortcut: $config->{groups}->{$last_group}.

That yields an arrayref that can also be shortcutted with the "pointer" operator to access individual elements, or using @{ ... } to access the dereferenced array, which is what you need for a push.

Hence (with typo's fixed):

push @{ $config->{groups}->{$last_group} }, $line;
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Perhaps a little commented Perl will make easier to understand:

use Data::Dumper;

my $last_group = "group_z";
my $line = "a line";

my $config = {          # a hash...
    groups => {         # ...of a hash...
        group_z => [    # ...of a list
            2, 3, "Peter", "Pan",
        ],
    },
};

push @{ $config->{groups}->{$last_group} }, $line;

print Dumper $config;
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Outputs:

$VAR1 = {
          'groups' => {
                        'group_z' => [
                                       2,
                                       3,
                                       'Peter',
                                       'Pan',
                                       'a line'
                                     ]
                      }
        };
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Have a look at perlref and perlreftut if you don't properly understand references yet, and then go on to read perldsc.

Update: Fixed a typo. Thanks ysth.