You want to open a filehandle, not an array of file names. You should print to a file handle, not to STDOUT.

use warnings;
use strict;

my $file2 = '/tmp/foo';

print "Enter filename to search\n";
chomp( my $file = <STDIN> );

if ( -e $file ) {
    die "File already Exists!\n";
}

unless ( -e $file ) {
    my @appfiles = glob "*";
    open my $fh, '>>', $file2 or die $!;
    print $fh "List of all the files in the directory:\n";
    print $fh "@appfiles\n";
    close $fh;
}
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See also the Basic debugging checklist