Modified your code a little bit:

1. Set variable option for your BrowseEntry, and that variable will hold the selection.
2. Completed your return_value function to print out the selection, so you can verify it.

Tested and worked.

use strict;
use Tk;
use Tk::BrowseEntry;

# Create the main window
my $mw;
$mw->{main} = MainWindow->new;

# Create the browse entry
my $v;
$mw->{a} = $mw->{main}->BrowseEntry(-variable => \$v, -label => "Selec
+t one:")->pack;
$mw->{a}->insert("end", "one");
$mw->{a}->insert("end", "two");
$mw->{a}->insert("end", "three");

my $frame = $mw->{main}->Frame()->pack();

$frame->Button(-text => "Ok",
                -command => \&return_value )->pack(-side => 'left');
$frame->Button(-text => "Exit",
                -command => sub{ $mw->{main}->destroy } )->pack(-side 
+=> 'left');

MainLoop;

sub return_value {
    print "return value is $v\n";
}
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