They probably ran an ad campaign to their App. Most of the pages have an app that they are using. So within their Fan Page they have this app within their Tabs. Alot of people start that way and provide Viral Content that encourages people to Like Share or Comment. If you put up a post then make sure to give a suggested Action to take.

Example - "Like" this post if you agree or "Like" if you agree or "Comment" if you don't

or something like that. It asks the people to take action for every post.

Damn, I was thinking it'd be a good idea to make a pinned post advertising a fan page since I don't have the money to purchase FB ads.

There is one tool that enables you to do that.
Auto Email Inviter for FB Groups Chrome extension.
This extension enables you to use any email list for inviting people that are not your friends to your group.
You can invite multiple email addresses from CSV or TXT files.

The best way is to have email list and then invitey email people to your fb group
I have application both that gathers emails and sends invitation by email on group.

Email harvester 45$
Invite by email application just 85$

If someone is playing a Facebook game, they will likely want to add more 'friends' who can help them out in game with freebies and will simply search via the built in Facebook groups search function for game groups where friends can be added. That's one reason these game groups have so many members.