When Analyse-it is installed for everyone who uses the computer, updating it replaces the copy everyone shares. Windows asks for administrator rights when the update starts. If you do not have them, ask your administrator to install the update; see Deploying the software.
Analyse-it asks you to close Microsoft Excel before it installs the update. Analyse-it starts again with Excel afterwards, and your license stays activated.
From the Administrator's Guide for Analyse-it 6.24.0: https://analyse-it.com/docs/user-guide/updating