Это массив хешей, попробуйте так:
$action->{'EVENTS'}[0]->{'CallerIDName'}
Подробнее см. perldsc .
Обновлено Пример, например:
use strict;
use warnings;
use Data::Dumper;
my $action = {};
$action->{'Events'} = [{'Muted' => 'something',
'Role' => 'something',
'Event' => 'something',
'Channel' => 'something',
'Talking' => 'something',
'UserNumber' => 'somenumber',
'CallerIDName' => 'somenumber',
'Conference' => 'somenumber',
'MarkedUser' => 'something',
'ActionID' => 'somenumber',
'CallerIDNum' => 'somenumber',
'Admin' => 'something'}];
#push hash into the array of hashes
push(@{$action->{'Events'}},{'Muted' => 'something',
'Role' => 'something1',
'Event' => 'something1',
'Channel' => 'something1',
'Talking' => 'something1',
'UserNumber' => 'somenumber1',
'CallerIDName' => 'somenumber1',
'Conference' => 'somenumber1',
'MarkedUser' => 'something1',
'ActionID' => 'somenumber1',
'CallerIDNum' => 'somenumber1',
'Admin' => 'something1'} );
for(my $i=0; $i < @{$action->{'Events'}}; $i++){
print Dumper($action->{Events}[$i]); #print entire hash in array index $i
#print callerIDName key(any key) of each hash
print Dumper($action->{'Events'}[$i]->{'CallerIDName'});
}