There are a couple of obvious ways. First the date is stored in a textvariable, so all you have to do is read $date. Second, if you read "perldoc Tk::ChooseDate", there is the get method.
Example->
#!/usr/bin/perl
use warnings;
use strict;
use Tk;
use Tk::ChooseDate;
#sets date thru the textvariable
my $date = '2006/6/6';
my $mw = new MainWindow;
$mw->geometry('200x50');
$mw->fontCreate('big',
-family=>'courier',
-weight=>'bold',
-size=>int(-18*18/14));
my $cd = $mw->ChooseDate(
-language =>'Italian',
-font =>'big', # the label font
-bg=>'lightsteelblue', #the label bg
-textvariable=>\$date,
-arrowcolor => 'black',
-arrowactivecolor=>'yellow',
-datecolor=> 'blue',
-dateformat => 1,
-orthodox => 0,
-daysofweekcolor=>'lightgreen',
-highlightcolor=> 'pink',
-linecolor=> 'green',
-yearcolor=> 'black',
-command=>sub{print "$date\n"},
)->pack(-fill=>'x', -expand=>1);
my $get_button = $mw->Button(-text=>'Get',
-command=>\&get)->pack();
my $cdcan = $cd->Subwidget('canvas');
$cdcan->configure(-bg=>'black'); # bg of weekdays bar
my $cdtop = $cd->Subwidget('toplevel');
$cdtop->configure(-bg=>'black'); # outline of popup
# sets the date thru set
$cd->set( y=>2005, m=>5, d=>5 );
MainLoop;
sub get{
print "getting....\n";
print "$date\n\n";
print "getting1....\n";
print $cd->get,"\n\n";
}