Windows 10 is far from a perfect operating system -- you may have recently upgraded only to find that you're experiencing a bug (or many). While I personally haven't experienced this, one of the more common bugs is a Start menu that refuses to open when clicked, or that frequently disappears from your desktop.
It's so common, in fact, that Microsoft has created a special troubleshooter to diagnose Start menu (and Cortana) problems.
If your Start menu won't open and you've tried the conventional fixes -- checking for OS updates (Settings > Update & security > Windows Update > Check for updates) and restarting your device -- it may be time to check out Microsoft's Start menu troubleshooter. The troubleshooter is a diagnostic cabinet file that you will need to download from Microsoft's support site. This link will take you directly to the download.
Download the troubleshooter and run the file -- a new window will open with a fairly simple walkthrough. After you run the troubleshooter, restart your device.
It's as simple as that -- hopefully you'll have no problem opening the Start menu once your device has been restarted. If you're still having troubles, Microsoft suggest creating a local administrator account (learn how to create a local account here) or, if necessary, resetting your PC.