Currently, users may geo-tag a tweet, but this feature is not enabled by default. Twitter is attempted to make money by selling advertising, and having more location info about more users will make Twitter's advertising inventory more value.
Users get the benefit of being able to see what is going on around them. This may be immensely useful for finding out about current news and further cements Twitter's position as the go-to source for news happening right now.
Imagine seeing a plume of smoke rising from ten blocks away - the first thing most of us tweeters would do is search Twitter for likely keywords, i.e. "smoke SF Financial district." Even the most artfully crafted searches still leave the user sorting through a huge amount of content. Twitter's new "nearby" feature will allow users to see tweets based on their geo-tagged location, making it much easier to see tweets being sent from the area of interest.
Note that geo-location enabled tweets have been possible for some time now. Several years ago, I worked with a developer who was making a "heat map" of tweets containing a given keyword. Higher tweet densities were represented with brighter colors on the map. I do not know the technical solution this developer had implemented, but so far as I can tell, Twitter's new API does not add new functionality, but rather, makes existing functionality available to users posting from 3rd party clients.
I am excitedly waiting to see the community's response to this new feature. New features often lead to new and previously un-imagined features, but the privacy challenges are daunting....
Good luck, Twitter!