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In Microsoft Edge click on the Hub the 3 lines icon then the Clock Icon (history) on right side top click clear history
You can also install ccleaner and ask it to delete the history. It can do much more besides so you need to go through and untick anything you don't want it to remove.
Did anyone read his question? He is asking how to set it to auto delete when you close edge. Like you can in firefox or IE. I am sure he knows how to clean his history manually. I would like to know how to do this as well.
Search Windows/Internet Options/Internet Properties/General/Delete browsing history on exit (must be check).
In Windows 10, click on the Windows icon in the toolbar, then choose Settings, then Network & Internet, the first option is for your WiFi settings, but if you look below under Related Settings you will see one titled Internet Options. Click that and it will open a box just like previous versions of Windows. Under the General tab you will find the Delete History on Exit. Something else they changed is the location of your cookie controls. They are still under the Privacy tab but now you have to click the Advanced button to see and set parameters for cookies. The Sites button still allows you to allow or disallow cookies for specific sites. Now for the disclaimer, I don't know if these settings apply to both IE and Edge or just IE since Edge also has a more limited settings function you can access when you are using it.
Hope this helps!