If the install button is "greyed" out on the Microsoft store app, you could be logged into the wrong Microsoft account on either the MS Store or the Insider Hub app.
By going to the Xbox Console Companion app you can log out of your Microsoft account which will in turn log you out of all other Microsoft apps. The next time you open the MS Store app or the Insider Hub app you will be prompted for your credentials, ensure you log into all apps with the Xbox Live account which has access to the Bleeding Edge technical alpha.
Update Windows 10
The Bleeding Edge technical alpha requires the latest updates for Windows 10 (specifically the may 2019 update).
You can use the Windows 10 update assistant to ensure your operating system had the latest software installed.