when adding the plugin as module in my page.

Are you displaying the module in a normal Joomla module position or are you trying to insert it into content using Joomlas loadmodule or loadposition usage? It looks like either a JS or CSS conflict. Are there any errors in your browsers developer console (press F12 then click the Console tab)?

Errors will be in red. Please provide any errors within the console. Seams like something is breaking the JS, which is typically due to a fatal JS error. The best way to insert CB Activity into Joomla articles is to use the CB Activity Bot, which is meant for doing exactly that. Review the description of CB Activity Bot within Extensions > Plugins for details on how to substitute streams into articles. If it's not there it may not be installed and you'll need to install it first, which is done automatically now when installing CB Activity.

Looks like your site maybe loading jQuery on every page. If this is the case try disabling CBs jQuery within CB > Configuration > Integrations and see if that clears up the issue. This usually happens due to the global $ variable being overridden by something other than a jQuery object.