Activity streams can be found on your homepage and in groups you have joined.
The 'Recent activity' stream on your homepage shows the most recent activity of all your groups, subscriptions, and bookmarks. This can include new questions asked by members or replies to your questions.
It basically provides you with one place to see all the updates.
If you scroll down the page, more activity will load and you can interact with this by liking comments, replying to them, or clicking on the question or blog to read more.
You can also find streams within groups, that only show that group's activity. These work in the same way.