A few updates back the Start Recording button was added so that users didn't have to change Settings > Broadcast Settings > Mode between Live Stream and File Output Only depending on whether they wanted to stream or record. Now you have both of those options on the main window.
For the button to work you need to go to Settings > Broadcast Settings, check the "Save to file" box, and define a location for the files to be saved. After clicking the Browse button and navigating to a folder you can either give a specific file name, or enter % for the file name to have OBS generate one automatically based on the date and time.