AP says that you need check Driving notifications and click "Add Driving shortcuts" under settings to be able to access the mode. However, you'll need to restart the app and poke around before those options even appear (see the image below to know what you're supposed to see). Besides driving mode, the updated Maps app adds an Audio toggle right on the navigation page that you can use to mute, unmute and switch sounds on for alerts only.
In addition, it adds a settings screen where you can curate what appears (such as your Google Photos) on your timeline. The update's currently being rolled out, and it might take a while to get it. If you don't mind dealing with the driving mode glitch that'll likely be fixed later, though, you can download the APK from Android Police right now.
Update (01/16/2016): A Google spokesperson told us that Driving Mode can only predict if you're going to the home and work destinations you programmed into your Google Maps app, as well as recently searched locations. It can't guess, say, if you're going to the mall or to any other place you don't go to every day...unless you work there, of course.
If you've preset home and work in Google Maps, you'll see ETAs at times when you're likely to be heading to these destinations. If you have location history enabled, we base these suggestions on location data, time of day and day of week. And if you're logged in, we'll suggest destinations based on recent Google Maps and Google searches when you open driving mode.
[Image credit: Android Police]