By default, print goes to the STDOUT filehandle (your terminal). To direct your output to a file, open a file for output, and print to the new filehandle. UNTESTED:

$size = 5000;
$overlap = 1;
$overlap_size = 0.00*$size;
$f = $ARGV[0];
local $/ = "\n>";
open $fh_out, '>', 'out.txt' or die "file out.txt: $!";
open(IN, "$f") || die "$f no\n";
while(<IN>){
    s/>//g;
    $id = ''; $annt = '';
    @ln = split /\n/;
    $defLine = shift @ln;
    $id = $1 if $defLine =~ /^(\S+)\s?/;
    $annt = $1 if $defLine =~ /^\S+\s+(.+)$/;
    $seq = join("",@ln);
    for(my $i=0; ($i*$size)<(length$seq); $i++){
        $subSeq = substr($seq,$i*$size, $size);
        $subSeq =~ s/(\w{60)/\$1\n/g;
        chomp $subSeq;
        $start = $i*$size +1;
        $end = $start + (length$subSeq) - 1;
        print $fh_out ">$id:[",$start,'-',$end,"] $annt\n$subSeq\n";
        if($overlap == 1 && ($i*$size+$overlap_size) < (length$seq)){
            $subSeq = substr($seq,$i*$size+$overlap_size, $size);
            $subSeq =~ s/(\w{60)/\$1\n/g;
            chomp $subSeq;
            $start = $i*$size +1+$overlap_size;
            $end = $start + (length$subSeq) - 1;
            print $fh_out ">$id:[",$start,'-',$end,"] $annt\n$subSeq\n
+";
            
        }
    }
}
close $fh_out;
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See also

• perlintro
• Basic debugging checklist