Twitter asking to "Authorize SNS to use your account"?
OK, I figured out what the app is asking. It's asking to create an account in the Settings > Accounts section on my Android device. This enables other applications, including stock applications and user-installed applications, to find out that I have a Twitter account and possibly use it to post to Twitter (provided I give consent to such apps to do so).

The following screenshot shows that after clicking the Allow button, the details pertaining to the user's Twitter account are stored on the Android device: