To open the built-in web browser or view recent downloads tap the downloads toolbar button:
If there are active downloads this button displays a badge indicating the number of downloads.
By default the web browser view will open. To download a link just tap the link, you'll be prompted to "Open" or "Download". If you choose "Download" the link target will be saved to the current folder.
To view or control active and recently completed downloads tap the "Downloads" tab bar button:
You can return to the browser at any time by tapping the "Browser" tab bar button.
Downloads also appear in the standard file list view, and can be controlled from there too. To return to the file list view tap the "Close" button at the top of the Browser or Recent Downloads view.
When Files Pro quits any active downloads will be paused. Next time you use Files Pro you can re-start the downloads again by opening the Recent Downloads view and tapping the pause button for each pending download.