First, if you have Touch ID enabled, none of the settings will show up at all, so first you have to go to Settings > Touch ID & Passcode to disable Touch ID. Don’t worry though, it’s only temporary. Once Touch ID is off, go to Settings > iTunes & App Store > Password Settings (it’s right under your Apple ID) and you’ll see a new section on that page called Free Downloads. If you haven't walked through these steps already, the option is likely green (for on) but also faded since you don't have access to change it.
New password settings in iOS 8.3. Why can't you change them?
There’s a toggle there that says “Require Password” and the text below it points out what the setting does. If you have it on, it says your free downloads will use the same password restrictions as purchases and In-App purchases (usually that means a password is required). If you turn that off, it says you won’t be asked for your password when you are downloading a free item.
Accessing the new password switch in iOS 8.3. Not grayed out! Ah, much better.
Now that you can see this option, you can go back to your Touch ID settings and re-enable it. You just have to have it disabled in order to get those settings to appear. After you turn Touch ID back on you can still go back and adjust the free app setting however you like.
It took some working through to determine how to get the setting to even appear, and if you have a “work phone” or there’s another reason your phone has a profile on it that allows it to be managed remotely, there’s a chance you won’t be able to change this setting at all. If you walk through those steps and it turns out you don’t get to change that setting, a security profile is a good reason why that could be the case.