Here's my go (based on examples in the docs)...

#!/usr/bin/perl

use strict;
use warnings;
use WWW::Mechanize;
use HTML::TableContentParser;

my $url = q|http://64.144.108.98/history/Allplayersstate.asp?State=AL|
+;

my $mech = WWW::Mechanize->new()
 or die "couldn't get Mech object: $!";
 
$mech->get($url)
  or die "couldn't get: $!";

my $html = $mech->content()
  or die "content failed: $!";

my $p = HTML::TableContentParser->new();
my $tables = $p->parse($html);

open my $fh, '>', 'tables.txt'
  or die "can't open output\n";
  
for my $t (@$tables) {
  for my $r (@{$t->{rows}}) {
    print $fh "Row: ";
    for my $c (@{$r->{cells}}) {
      my $data = $c->{data};
      print $fh "[$data] ";                          
    }                         
    print $fh "\n";                       
  }
}
[download]

Extract from output:

Row: 
[13507] 
[4/30/1998] 
[<a href="Phistory.asp?Pid=13200">Acoff, Fred</a>] 
[1678 (1678)] 
[AL] 
[<a href="Tall.asp?Tid=785&SBy=LastName+Asc">3/16/1997</a>] 


Row: 
[71636] 
[5/31/2007] 
[<a href="Phistory.asp?Pid=56179">Akhtar, Jibran(Brian)</a>] 
[152 (152)] 
[AL] 
[<a href="Tall.asp?Tid=3332&SBy=LastName+Asc">7/8/2006</a>] 


Row: 
[45757] 
[12/31/9999] 
[<a href="Phistory.asp?Pid=30169">Alexy, Tom</a>] 
[1342 (1475)] 
[AL] 
[<a href="Tall.asp?Tid=3232&SBy=LastName+Asc">3/25/2006</a>] 


Row: 
[57249] 
[3/31/1991] 
[<a href="Phistory.asp?Pid=37455">Alford, Josh</a>] 
[594 (594)]
[AL] 
[<a href="Tall.asp?Tid=0&SBy=LastName+Asc"></a>] 


Row: 
[57242] 
[3/31/1991] 
[<a href="Phistory.asp?Pid=37448">Alford, Tom</a>] 
[652 (652)]
[AL] 
[<a href="Tall.asp?Tid=0&SBy=LastName+Asc"></a>] 


Row: 
[70273] 
[3/31/2009] 
[<a href="Phistory.asp?Pid=52193">Alter, Torin</a>] 
[1289 (1340)] 
[AL] 
[<a href="Tall.asp?Tid=3356&SBy=LastName+Asc">9/10/2006</a>] 


Row: 
[3865] 
[5/31/1994] 
[<a href="Phistory.asp?Pid=3673">Alvey, Clyde Henry</a>] 
[1578 (1578)]
[AL] 
[<a href="Tall.asp?Tid=0&SBy=LastName+Asc"></a>] 


Row: 
[55518] 
[7/31/1990] 
[<a href="Phistory.asp?Pid=35871">Ammons, Jermey C.</a>] 
[859 (859)] 
[AL] 
[<a href="Tall.asp?Tid=0&SBy=LastName+Asc"></a>] 


Row: 
[59871] 
[4/30/1993] 
[<a href="Phistory.asp?Pid=39893">Anderson, Derriel</a>] 
[1173 (1173)] 
[AL] 
[<a href="Tall.asp?Tid=0&SBy=LastName+Asc"></a>] 


Row: 
[57295] 
[9/30/1990
[<a href="Phistory.asp?Pid=37497">Ardoin, Jean Louis</a>] 
[1567 (1567)] 
[AL] 
[<a href="Tall.asp?Tid=0&SBy=LastName+Asc"></a>]
[download]