I downloaded the logs file but am having a hard time opening it with various extractors.

I set up a cron job on my hosting server using a wget command to initiate this url on the 28th of every month at 3am:

http://mywebsitehere.com/index.php?option=com_comprofiler&task=pluginclass&plugin=cbpaidsubscriptions&cbpact=cbsubsclass&cbpclass=mailer&user=62&cbpmailerdo=cronprocesssendmailqueue&key=keyinforemovedforsecurity&format=raw

I logged into the backend of the site this morning (May 29th) and there were no new "emails sent".

So I did a manual "create list and send emails" using the link above. That sent the emails.

However (this is off topic) - using the shortcode in the body of the email for [SUBSCRIPTION_EXPIRATION_DATE] did not work. My email message says this instead: "Your membership with the MOB expired on [SUBSCRIPTION_EXPIRATION_DATE]. We have a bunch of new events lined up for 2013! So won't you please renew your membership and pay your annual dues ..." 

So how does this cron job really work - since the site requires an admin to log in - how can a cronjob initiate a URL without a logged in user?

I thought the whole purpose of this mailer and cron job ability was so that I don't have to log in every month and create new mailers, build a list queue and then send the emails?