Anything you need another company's permission to access doesn't belong to you. Google could lock you out of your own email at any point. Any important messages you've sent or received over the years can be taken from you and you could be left with no means at all to get them back. Having a local copy of your correspondence is great for piece of mind.

Because thunderbird stores messages in MBOX format your messages can be read by anything that can handle text files and you can use tools like grep to search through your messages very quickly from the command line. If you get an address with a new email provider having a copy of all your messages on your hard drive means that you can easily move them to the new account.