First, open the website you want to pin in Internet Explorer. Tap the More (…) button and tap Pin to Start in the menu that appears. Windows Phone 8.1 supports live tile updates from websites that have configured the feature, just as Windows 8 does.
Thanks to www.howtogeek.com for this information!
If you have another type of smartphone or tablet, it probably has this feature, too. Just open its browser and look in its menu for an option named something like “Add to home screen” or “Pin to home screen.”