Twesents is another way to keep you using Twitter and playing nice with your virtual friends by sending virtual gifts.
Now the confusion. There is no help so I walked the screen. It simply asks you to choose a twesent. But there was an amount of credits below each twesent you choose. No mention of how you get/earn/find credits at all. Then it immediately jumps to the next screen and asks for your Twitter username and password. I am not saying it is not safe, just not well described.