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 save the file to your device first.
To save one of these file types to your device, after selecting to download it look for the blue down arrow icon in the top right of your Safari browser.
Tap this icon 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.