Just as the recording of your Tour, the voice navigation is based on GPS. That's why you can use it even without internet connection. To make your Tour available for offline usage, select it from the list of Planned Tours in your app. (this only works with planned Tours!)
Now move the toggle "Store for offline use" to the right. Tour and maps will now be saved on you mobile device. Once the download is completed your Tour is available offline.
Please note: To plan new Tours, you still need to be connected to the internet
Avoiding Roaming Charges
To make sure komoot and other apps don't access the internet while you are abroad, check your phone's roaming settings
Android: Deactivate in your phone's settings -> Mobile Data -> Data roaming
iPhone: Deactivate in your phone's settings -> Mobile -> Data roaming