As others have already explained, you have to write the data back to the file, either by using in-place editing or just manually doing it yourself:

use strict;
use warnings;

my $file = shift;
open FILE, $file or die "Can't read from $file!\n";

my @lines;
while (my $line = <FILE>) {
    if ($line =~ /\>/) {
        push @lines, (split( /\s/, $line ))[0] . "\n";
    } else {
        push @lines, $line;
    }
}
close FILE;

open FILE, '>', $file or die "Can't write to $file!\n";
print FILE @lines;
close FILE;
[download]

use strict;
use warnings;
use Tie::File;

my $file = shift;
tie my @lines, 'Tie::File', $file or die "Can't read from $file\n";

for (0 .. $#lines) {
    if ($lines[$_] =~ /\>/) {
        $lines[$_] = (split( /\s/, $lines[$_] ))[0] . "\n";
    }
}
[download]

L-LL-L--L-LL-L--L-LL-L--
-R--R-RR-R--R-RR-R--R-RR
B--B--B--B--B--B--B--B--
H---H---H---H---H---H---
(the triplet paradiddle with high-hat)