According to the man page, Net::SMTP debugging is sent to STDERR (see the parent class Net::Cmd documentation.) Since you're using Apache, the STDERR stream will go to the normal Apache error log file, perhaps /var/log/apache2/error.log.

If the above is not what you want, and you truly want to "capture" the output, for further processing by your script, you'll need to redirect STDERR to a file. The open function documentation shows how to save and restore STDERR.