Why do I need a functional Interface to work with lambdas?

The most important reason why they must contain only one method, is that confusion is easily possible otherwise. If multiple methods were allowed in the interface, which method should a lambda pick if the argument lists are the same ?

interface TestInterface {
    void first();
    void second(); // this is only distinguished from first() by method name
    String third(); // maybe you could say in this instance "well the return type is different"
    Object fourth(); // but a String is an Object, too !
}

void test() {
    // which method are you implementing, first or second ?
    TestInterface a = () -> System.out.println("Ido mein ado mein");
    // which method are you implementing, third or fourth ?
    TestInterface b = () -> "Ido mein ado mein";
}

You seem to be looking for anonymous classes. The following code works:

public class Test {

    public static void main(String...args) {
        TestInterface i = new TestInterface() {
            public void hans() {
                System.out.println("Hans");
            }
            public void hans(String a) {
                System.out.println(a);
            }
        };

        i.hans();
        i.hans("Hello");
    }
}

public interface TestInterface {
    public void hans();
    public void hans(String a);
}

Lambda expressions are (mostly) a shorter way to write anonymous classes with only one method. (Likewise, anonymous classes are shorthand for inner classes that you only use in one place)

When you write :

TestInterface i = () -> System.out.println("Hans");

You give an implementation to the void hans() method of the TestInterface.

If you could assign a lambda expression to an interface having more than one abstract method (i.e. a non functional interface), the lambda expression could only implement one of the methods, leaving the other methods unimplemented.

You can't solve it by assigning two lambda expressions having different signatures to the same variable (Just like you can't assign references of two objects to a single variable and expect that variable to refer to both objects at once).