Is something going on w Twitter? It kicked me out and asked for my phone #

[better]

7. Most likely it is to do with 2-factor authentication.

It's a security measure, not to protect against bots, per se, like the pictures, but to protect against your account being hacked/hijacked. When a login is detected from an unrecognized device, many systems will send a text message to your phone, containing a verification number, which you can then provide back to the system to verify that it was, in fact, you logging in from the unrecognized device.

I can certainly understand if you are unwilling to give out your phone number, but just for the sake of information, this is why it's often required.