Unfortunately there's no automatic way to move the data from Chrome application to new native application. This has o be done manually by exporting the date to file (or to Google Drive) and then import it in the new app.
Let's start with installing Advanced REST Client native application if you haven't done that already.
After new application is installed go to ARC Chrome application and export the data.
In the top right application menu select Import and export data.
Now open the new Advanced REST client native application. In the application menu select File > Import data.
Similar to previous application the import / export panel is now displayed. Select Open from file to read exported file.
You can still use the old application but to transfer newly created data to the other application you will have to repeat this procedure.