Well, you aren't doing anything after spawning the process, so the script is ending and killing the process. Try the following:
$Expect::Debug = 1;
my $string= '/usr/bin/ssh 127.0.0.1';
my $exp=Expect->spawn($string) || die "error $!\n";
$exp->expect(2, "password: ");
This works fine on my system.
If your goal is to interact with the ssh session after scripting the login, try adding:
after you have finished logging in.