Phones are annoying, and the UI of most apps is user-hostile. Here are the equivalent steps you have to do to accomplish the same thing as this button does:
- Wipe your hands from moisture/whatever you had on them.
- Find your phone in your pocket.
- Orient it.
- Unlock it.
- Exit whatever app you left running in the foreground.
- Open the app drawer.
- Find the ordering app.
- Click through several layers of UI to get to the right product.
And only now you're in the same position as after just pressing a button. This is a significant cognitive load. When I realize I need to order something, I want to order it, not navigate through countless steps. Having to do more than two actions is too much.
Exactly, a cellphone app can get the job done but it is a great cognitive deviation from whatever task you had at hand at the moment you realize you ran out of something(1). So amazon's dash button is not an innovation on getting something done, but get it done faster and more conveniently, and that alone makes sense for an enterprise whose profit depends greatly from understanding consumer behavior. So I agree with TeMPOraL, it is not ridiculous at all.
And there's a bigger picture outside of comparing the convenience between using this little button as opposed to mobile apps, but that's off topic for this branch of the conversation, if you're interested IMHO & my brief reasoning, read this other brach[2] (also because DRY).