Perhaps eight or ten years ago I had a Polish girlfriend and visited her in Poland and lived with her friends for a bit. I noticed how they mainly used the Internet. They had an ICQ-like polish IM (GaduGadu) running and they ONLY changed the status messages on that thing to inform people what they were doing.

I found it amusing at that time. Later I saw Twitter, and it was very similar.

So yes ,it does solve a problem: Broadcast short pieces of information to friends without having to start a conversation.

At least I think that was the original idea.

It allowed people to update their social status pretty much from anywhere at any time with minimum friction.

As far as I can tell, Twitter was built on the idea that (some) people want to get advertised to by the brands (corporate or personal) they admire. Until Twitter came along, the "best" way to do this was basically through giving over your email address and waiting who-knows-how-long for a who-knows-how-lengthy newsletter, and hoping they don't sell your contact information to spammers in the meantime. Twitter enforces brevity, which encourages timely little facts rather than big, polished PR-jobs, and it's a pull-based, rather than push-based model, meaning that you can always unsubscribe from anyone whose channel becomes more noisy than signally.

Twitter boomed primarily because its target early-adopters were precisely the kind of people who already had lots of followers to advertise to (celebrities, reporters/bloggers, "personalities", etc.), and who wanted a more streamlined, more informal, lower-expectation medium to push out their little updates. Twitter was such a channel, but although they were on it, their users weren't—so, since they were all of the getting-the-word-out bent, the next thing they all (simultaneously) got the word out about was Twitter itself. If not for this clever bit of social engineering, Twitter would never have become the "thing that comes after Facebook in 90% of sentences about social media."

Twitter indulges people's need to feel like they're being heard. Giving people a voice is a very powerful tool.

"What pain did Twitter address?"

It allowed people to keep up to date with people and services they cared about in a format that they could quickly consume.

Wanting to send group emails to people, but not wanting to burden their privacy, and trying to get more friends in the process: getting social proof. Also, tying it in with escaping the rss reader eco system: news pushed to you. Also, making web publishing simple: self-expression for plebs.