WordPress will automatically download the update, unpack it, verify it, install it, upgrade the database (if necessary) and tell you it’s worked OK. You can then see the About WordPress screen.
If anything goes wrong you may have to revert to your backup. Fingers crossed that never happens.
Every now and then a new version of WordPress becomes available. Applying the update to your website should be as easy as clicking on the Update Now button. But you should ALWAYS take a complete backup before applying changes. You never know when something might go wrong.
After applying updates use the Check again button to force a new check for different versions of plugins and themes.