The option to download the .fit file of any activity synchronized with Suunto app is available in the upper right menu of the training.
Go to your Diary (second icon on the lower access bar) and on the Training view.
Select whichever training you are interested in exporting the file from.
Press the upper right menu (three-dotted line).
Select the Download FIT file in the dropdown menu.
If you haven’t already, allow Suunto to access photos, media and files on your device and then the download will start. The file will be ready and available in your files.