Quick notes keep popping up on my iPhone

In the last week or so, whenever I start a text message to my wife, a little white box pops up in the lower right corner with her name and the Message icon (the little green speech bubble). If I swipe to to the right, it collapses, but there's still a little grey bar there with an arrow on it.

The only way to get rid of it is to click on the little white box which opens a new note with her name at the top. I can then swipe up to to get rid of it, just like any app.

What is this? Why is it doing it all of a sudden? I'm not touching anywhere on the screen to make it happen, it just does... very annoying!

Does anyone know what it is and how to turn it off?