What does it mean to "subscribe" to an IMAP folder?

This is due of the peculiarities of the IMAP protocol. Strangely enough, the authors of that protocol not only thought about email when they dreamed it up, they also thought about usenet and newsgroups. The newsgroups of the usenet form a large hierarchy with thousands and thousands of groups. The original aim of IMAP was to support having the complete hierarchy available through your IMAP account. Of course, you wouldn't want all those groups to show up in your mail client, thus the concept of folder subscriptions was added to the protocol.

If I remember correctly, Thunderbird also has an option to simply show you all your available folders without going through the motions of subscribing to single folders. It has nothing to do with offline use or synchronization.

If you want to know more about the IMAP protocol, check out the corresponding Wikipedia article. You can also try to read the corresponding RFC which makes up the protocol and which is full of examples concerning usenet newsgroups (and has only a few examples that relate directly to email).

The folders you subscribe to are the ones displayed by Thunderbird. It subscribes every folder it finds when you first setup the account and any folder you add through Thunderbird is automatically subscribed as well. I guess that option is there so if you were to add a folder through another application or a webmail-ish app and you wanted that new folder to be displayed as well you can just turn it on.

Maybe this link will clarify things. Here's a quote:

Your IMAP email client (eg Thunderbird) will display the folders in your account to which you are "subscribed".

Generally, this will be all of the folders in your account, and only the folders in your account. However, in some situations you may find that there are folders in your account that do not appear in Thunderbird, or folders that appear in Thunderbird that are not in your account. This can be fixed by subscribing to (or unsubscribing from) the offending folders.

There are already a couple of very good answers to this question (which I’ve upvoted) but I thought I’d complement them with an answer detailing the relevant commands specified by the IMAP protocol, RFC 3501.

Subscribing to folders

In IMAP, subscriptions are used as a way of marking which folders should be displayed by IMAP clients. Some system folders may provide functionality but not contain emails that are of interest to the user, e.g., an Exchange server contains folders such as Contacts, Journal, Sync Issues).

Two IMAP commands are used for listing IMAP folders:

• the LIST command lists all folders.
• the LSUB command lists only subscribed folders.

The output of the LSUB command is affected by 2 commands:

• SUBSCRIBE adds folders to the list displayed by the LSUB command.
• UNSUBSCRIBE removes folders from the list displayed by the LSUB command.

See What is the use of Subscribe and Unsubscribe commands in IMAP4? for examples of the effects of the subscription commands.

Note also that folders are marked as subscribed on the IMAP server so that all clients see the same list of folders. See Is IMAP (un)subscribe meant to work across mail clients?.