I don't trust 3rd party apps with my password, but I'm curious about what it does once it has it? Spam tweets? Connect with your friends? etc...
The game does not auto-tweet. All the lines of text in the game are under 140 characters, and you have the option to tweet each one if you feel like sharing. Please keep in mind that your friends might not be interested in seeing the whole game in their twitter feed. Tweeting is completely optional and does not affect your game progress in any way.
This link provides a lot of info: http://geraldonascimento.net/sworcery-how-4-four-guys-managed-to-hijack-the-world/
The game has flash flooded and managed to get into #2 yesterday on the top charts. Because of this twitter integration.
Full disclosure: I'm the author of that post. It's my best explanation for why Twitter asks for the password.
There are two real answers:
1 - Technical Answer The technical answer is that a twitter application needs your password so it can register itself and tell Twitter.com that YOU trust this application, in this case Superbrothers: Sword & Sworcery EP.
2 - Philosophical Answer The philosophical answer is that asking you for your password is a way of knowing if you trust the game completely.
My personal opinion: you should.
EDIT / UPDATE:
I have since then played the game, and he asks you for your username and password. If you are not used to Twitter clients, they all work the same way:
1 - They ask you to login to Twitter. 2 - Twitter asks that you allow said app to access your Twitter feed and post stuff there.
Sword & Sworcery is sorta like that. Cept its not a Twitter client. They handle that process themselves.
The game is very tightly integrated with Twitter. Watch the #sworcery hashtag to get an idea of what it's going to be sending out. (And to see classic Twitter users feuding over whether it should/should not do so.)