The App Store knows what you have already installed.
Keep in mind that the App Store is NOT a repository for your apps, it's an online store, where you purchase those apps. If the app is already installed, it will let you know.
If you need to reinstall that app, then uninstall it first. The Launch Pad can be a good method to uninstall your app. That will work very neatly, and then the App Store will let you install your app again. You should try to log out, then log back in to the App Store. If you want to keep a copy of the app for a later use, then back up the app to another drive.
If for some reason the App Store still does not let you install again, and the support info available from the Quick Links in the App Store does not help you, then there may be something gone wrong with your AppleID or your Apple account, etc. You should contact AppleCare for some help with that.
You could also log in to another Mac, using your AppleID, which should show those apps as available to install or download.