You need to be a Jira Administrator in order to install Automated Release Notes.
First, login as the Administrator to your Jira instance and select "Apps" from under Jira Administration. Jira Administration menu is available in the top right corner on click of the 'Cog Wheel' icon.
In the "Find new apps" page, search for "Automated Release Notes".
Click on the "Free trial" link on the right side of the corresponding listing.
This will result in the Automated Release Notes app being installed and activated.
A confirmation dialog will tell you that the installation was successful.
Installation steps remain same for both, Server/Data center as well as Cloud version of the app. There might be minor variations in different Jira versions, but the overall procedure is similar to what is followed in managing other apps.