Beside other answers, I think the main reason to "why using Functional Interface other than directly with lambda expressions" can be related to nature of Java language which is Object Oriented.
The main attributes of Lambda expressions are: 1. They can be passed around 2. and they can executed in future in specific time (several times). Now to support this feature in languages, some other languages deal simply with this matter.
For instance in Java Script, a function (Anonymous function, or Function literals) can be addressed as a object. So, you can create them simply and also they can be assigned to a variable and so forth. For example:
var myFunction = function (...) {
...;
}
alert(myFunction(...));
or via ES6, you can use an arrow function.
const myFunction = ... => ...
Up to now, Java language designers have not accepted to handle mentioned features via these manner (functional programming techniques). They believe that Java language is Object Oriented and therefore they should solve this problem via Object Oriented techniques. They don't want to miss simplicity and consistency of Java language.
Therefore, they use interfaces, as when an object of an interface with just one method (I mean functional interface) is need you can replace it with a lambda expression. Such as:
ActionListener listener = event -> ...;