Why app connect to internet with local file?

In order for the WebDAV server to work, Textastic sets "UIRequiresPersistentWiFi" to YES in its Info.plist file.

This makes sure that a Wifi connection is enabled if possible. If you are in Airplane mode, this results in a system popup that asks you to connect to a Wifi network. You can just close that.

Textastic doesn't actually connect to the internet when you start it. Connections are only started when you connect to a remote server.

You can read more about UIRequiresPersistentWiFi in Apple's developer documenation at: https://developer.apple.com/documentation/bundleresources/information-property-list/uirequirespersistentwif

This was required in earlier versions of iOS to be able to start the WebDAV server and bind it to the local Wifi ip address. 

I need to check if setting UIRequiresPersistentWiFi is still required in the latest version of iOS.

I can see how the system prompt that is shown in Airplane mode might be misleading.

thanks a ton for that!

Is WebDAV server runs on open of Textastic? If yes, possible to manual start only?

No, the WebDAV server is only started manually. See Settings -> WebDAV server.