You will need a version of Eclipse where you can add new plugins. This generally means a copy that you own and that isn't shared. This should be a relatively recent version, say from the last 4 years.
From within Eclipse, go to install new software (generally on the help menu). This brings up a dialog. Choose to add a new site and enter the URL https://www.cs.brown.edu/people/spr/bubblesupdate as the site.
When you next refresh the list with this site (or with all sites), you should find an entry for the Code Bubbles plugin.
From the install new software menu, choose the Code Bubbles plugin and have it installed into Eclipse. You will probably have to restart Eclipse for this to have an effect. Note that this does not install Code Bubbles, only the plug-in.
Once the plug-in is running with Eclipse, there should be a Code Bubbles menu. This menu should have one button "Install Code Bubbles". (It will change once Code Bubbles is installed.) Clicking on this will start the Code Bubbles installation. Code bubbles will be installed into the Eclipse directory (varies by machine) in the subdirectory bubbles. (This is generally the same directory that contains the plugins and dropins folders.)
Once installation is finished, the menu button will change to "Start Code Bubbles". (This is also how it will appear when you next run Eclipse as long as Code Bubbles is installed. Note that if you do a traditional installation of Code Bubbles, the start button should still be there and should still work.) This will bring up the Code Bubbles interface. You can then minimize the Eclipse window and use Code Bubbles exclusively (or go back and forth).
If you want to run Code Bubbles standalone at this point you can. Find the bubbles installation directory inside Eclipse. Then create a script using your favorite shell that does java -jar of te bubbles.jar file in that directory. These is also a preference option to automatically start Code Bubbles whenever Eclipse is started that may work. If you are going to run Code Bubbles standalone, you probably want to reenable auto_update (preferences option or in the System.props file of ~/.bubbles).
Note that this process is relatively new and not completely tested (or understood). Please report any problems or suggestions to Steven Reiss.