Here are some comments to your code:

• You don't need to open the input file before passing it to $extor->parse_file
• Use the glob function (glob("./*.html)) instead of readdir and parse the resulting files for the ".html" extension
• You are not extracting the links (you are not using the method $extor->links)
• You are using $_ inside the foreach loop, but that variable is not initialized (because you are using a "loop variable" ($file))
• $file is declared outside the foreach loop, you should avoid this and restrict the scope of the variable at the minimum
• You should get the habit of closing the filehandles

Here is a version of your code with all these issues corrected (untested):

use strict;
use warnings;
use HTML::SimpleLinkExtor;

my $FileOut = "output.txt";

open FileDOut, ">$FileOut" or die "Can't open output file $FileOut: $!
+";
print "Summary written in file $FileOut\n";

while (my $file = glob ("*.html")){
  print FileDOut "$file\n";
  my $extor = HTML::SimpleLinkExtor->new();
  $extor->parse_file($file);
  my @links = $extor->links();
  print FileDOut "$_\n" for (@links);
  print FileDOut "\n";
}

close FileDOut;
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I don't think you need to pass a file handle to the parse_file accessor,passing a filename will do. Then you need to use the links accessor to get all the links in the htm file -

foreach $file (@files) {
    if ($file =~ /^(.*)htm/i){    
        print FileDOut "$file\n";
        my $extor = HTML::SimpleLinkExtor->new();
            $extor->parse_file($file);
        my @all_links   = $extor->links;
            print FileDOut join "\n",@all_links;
    }
}
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As above... And I doubt that parse_file wants the entire file passed to it as an array of lines ! Which is what:

    $extor->parse_file(<$fh>);
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