You will get a small banner across your playing screen alerting you to the fact a visitor has arrived!
That banner will vanish, and on the right side of your playing screen there will be a countdown timer (60 minutes per butterfly) and a small picture of that butterfly!
The butterfly will also show up in your forest, allowing you to use it's ability once!
When you click on the timer, and the small picture of the visitor, it will show you some awesome information, as well as the option to purchase one egg! It will also state how much the egg will cost you.