Either something was installed on your phone, without you knowing, or you added log in info to this site when you went there. Giving them access to your account on Twitter, hence the posts.
You need to get with Twitter if you want to change permissions, however I don't see that fixing it.
You need to scan your phone for viruses, bad apps, malware, etc. Then, once cleaned, change your passwords for everything you access on your device. Starting with your Twitter account. I say do them all because it is possible that this could escalate to accessing other things on your phone.