You can find a million different marketing blogs on "why using hashtags is imperative to your Twitter success." But you should think of this list as more of a "why you should use hashtags for your Twitter enjoyment."
Find Like-Minded People
For example, let's say that you are super into Doctor Who, but all of your friends think British TV is weird (true story). You don't have anyone to talk to about the new doctor, John Hurt, or the epic 50th Anniversary Special (maybe coming soon to a theater near you!). What do you, as a super fan, do? Well, you can take to Twitter, and because of hashtags, you can easily find people who are geeking out about these very things.
Here is a personal anecdote to back up this sentiment: I don't have any real life friends that use Twitter. All of my actual friends spend their time on Facebook. So when I became interested in social media almost a year ago, I started my Twitter with zero followers. I don't have a personal business, or any real need for a lot of followers, so finding followers has never been a goal for me. However, I have noticed an uptick in people who follow me when I make several posts a day with hashtags.
Support A Cause
As an example, I am currently very into National Novel Writing Month - or #NaNoWriMo. In November, people all around the world band together while trying to write 50,000 word novels. It's a pretty lofty goal, but it forces people to sit down and tell their story. One of the best ways to get the word out, is to use the Twitter hashtag. It shows my support, all of my followers can learn what it is, and if enough people talk about, it will start trending. So, every time I tweet about my novel (a real work in progress), I try to add the NaNo hashtag.
And while NaNoWriMo is a great example, you can also support your favorite store (think #maxinista) or social movement (think #occupy). It's just one way to let people know how you feel about a cause, product, or event.