You call a subroutine that prints something to the default output handle, and returns 1 (because the print was successful). Then you write that 1 into FILE, which is opened to 'files.txt'.

If you want the string "How are you?\n" to be end up in the file, you need to write

print FILE "How are you?\n";
[download]

Or something along the lines of

open my $file, '>', 'files.txt';
select $file; # print to $file by default
doit();
close $file;
select STDOUT; # print to STDOUT again by default
[download]

    sub doit{
     return "How are you?\n";
    }
    open (FILE, ">files.txt");
    print FILE &doit;
    close(FILE);
[download]

    sub doit{
     my $handle = shift;
     print $handle "How are you?\n";
    }
    open (FILE, ">files.txt");
    doit(*FILE);
    close(FILE);
[download]

A simple way to correct your sample code would be to replace:

with:

Now, your subroutine is returning a string, so your print statement below will print the string itself rather than the results of your successful print statement.

sub doit{
 print "How are you?\n";
 $x=1+1;
 print "$x\n";
}
open(FILE,"<monkss.pl");
@lines=<FILE>;
close(FILE);
open (FILE, ">files.txt");
$flag=0;
foreach $line(@lines)
{
 if($line =~ /doit\{/)
 {
  $flag=1;
 }
 if($flag)
 {
  print FILE $line;
 }
 if($line=~ /\}/)
 {
  $flag=0;
 }
}
close(FILE);
[download]

sub doit{
 print "How are you?\n";
 $x=1+1;
 print "$x\n";
}
[download]

Or, similarly:

#!/usr/bin/perl

use warnings;
use strict;

sub doit {
    print "How are you?\n";
}

seek DATA, 0, 0;

while (<DATA>) {
    print if /^sub doit {/ .. /^}/
}

__DATA__
[download]

If &subroutine is antiquated Perl, what's the preferred way to reference a sub?

Have you read the most basic documentation, perldoc perlsub?

And if that's insufficiently clear, try the subroutine section of Tutorials ...for the content itself, and for the experience of trying to answer your own questions.