First of all, we're not asking you for your iCloud login/password, 2Do is
Pardon the pun but that's an important point. We don't get to ever know about your login details. Those are saved securely within the application.
Secondly, you're referring to iCloud's automatic document syncing mechanism. 2Do does not utilize iCloud's document syncing feature. We've explained this elsewhere on the forum but in short that would require a complete re-write of the application and would demote it to the level of simplicity the Reminders app has to offer.
2Do instead supports CalDAV Sync. Here's an explanation of CalDAV on wikipedia: http://en.wikipedia.org/wiki/CalDAV
CalDAV allows any client to be able to synchronize against an online calendaring account. This is what iCal, for example, uses to sync your data back and forth between iCloud and your Mac. 2Do is just another 'CalDAV Client' which now supports MobileMe, iCloud, Custom hosted iCal Servers and Yahoo! Calendar - all of which are synchronized using the CalDAV protocol. If you for example log into www.icloud.com
you'll see there's an online version of your calendar - that's what 2Do synchronizes against and requires your account details in order to connect and retrieve data.
We hope this answers your question. If you've got more feel free to ask.