Interestingly enough, it's not exactly a new feature, Twitter enabled users to do the very same thing up until a couple of years ago when it dropped the feature saying that not a lot of people used it and it took too much of a toll on the servers.
Twitter introduced several new features and changes last week. The "following" page has been revamped. This page lists all of the Twitter users someone is following, but it now has two tabs, a "People" and a "Tweets" tab.
The People tab has biographical information, but the Tweets tab, as the name suggests, lists all the tweets that particular user sees at the moment.
The new feature could prove useful, seeing the actual tweets from the people someone follows rather than just a list of names, makes it easier to take an informed decision and stumble upon an interesting account.
This feature is not new though, a very similar one was killed off in 2009. At the time, Twitter said it was rarely used, so it may be that it won't see much use even now, especially since it's rather buried.
Twitter also introduced a similar feature in a much more visible place, the shuffle button which you'll notice in top right corner. It enables you to jump to the Twitter of any of the people you follow.
Again, the feature is designed to drive up engagement and make it easier for users to find accounts they may want to follow. Considering that these are people followed by the ones you're following yourself, there's a good chance you'll find someone of interest.