Thank you for contacting us, and I'm sorry for the issue you are experiencing. You're not missing something here. When you tap the "Email Backup" option, you should see a dialog pop up telling you about the email backup process. Unfortunately, I don't know what's happening here, because it's working correctly on my iPhone, and we haven't see other reports of this behavior since the v6 release.
Do you by any chance have mSecure installed on another device, and is your informations synced over to that other device?
While I don't know what's causing this to happen, I'm hoping a reinstallation of mSecure will fix the problem. As long as you are able to see all your information in mSecure on your PC, go ahead and delete mSecure from your iPhone and then install it again from the App Store app. After it's installed, sign in to your account.
After setting up the app again, before you perform a sync with your PC, go ahead and see what happens if you tap the Email Backups option in the Settings. Did the dialog pop up? If so, go ahead and sync your data over from your PC, then try tapping the Email Backup option in mSecure's Settings again. Did the dialog pop up again?
Something is wrong on your iPhone, but I'm not sure what it is. Especially with the app in a fresh state, you should not have any problems creating an email backup. I don't think you need to give mSecure permission to use an email app, so the only thing I can think of is that you might be using a different email app that's not getting the message that mSecure is needing to use it to send an email. Are you using Apple's Mail app on your iPhone, or have you set a different "Default Mail App" in iOS Settings?
This might not be critical, as you may have your data stored in the mSecure Cloud for syncing (or a different cloud service), and automatic backups are stored in your iCloud Drive account, but the feature should still work. The fact that it doesn't indicates there is some type of problem, so it's always good to get these things either solved or at least know why it's happening.
In the Settings of mSecure, you'll see a "Backups" option. Tap that option to see if you have any backups stored in your iCloud account. Do you see any backups in the "Backups" screen? Also, go ahead and tap the "Backup Now" button at the bottom of the screen. Does a new backup created in the list, or does it seem like nothing happens?