For some file types (like PDFs, images, text) your device will be able to open them directly, but for other file types (e.g. EPUB, MOBI, ZIP), you will need to open them from your Downloads folder.
To do that, after selecting to download look for the blue down arrow next to your browser's address bar.
Tap this icon followed by Downloads...
And you should see your file listed. Then tap the file that you’d like to open and it should open the file in the appropriate app; e.g. Apple Books for an EPUB file, the Files app for a ZIP file.