When you click the Profile menu then leaf will show you the profile for the ancestor listed at the top of the application:
Likewise you can click the Profile menu in a popup to show the profile for the person you have selected on the ancestor or descendant chart.
The profile shows you the information stored in FamilySearch for this person, and the button indicates where a person, date, place, or ordinance is missing.
The button indicates that the family has only one child. leaf can't tell whether the family has more children, but it is frequently the case that families have more than one child, so this is marked as a possible place where additional research is needed.
The button will open a new tab in your browser and visit the person shown in the profile on the New FamilySearch web site. You can edit information about the person here. Note that you may need to login first.