You are just defining the type of person here (the colon stands for type annotations eg:propertyName:Type) :

person: {
    firstname: string;
    lastname: string;
}

What you have done here is telling the compiler that person is an object literal with 2 string properties firstname and lastname. It defines what can be store in that property, not what it contains.

You should assign a value first, otherwise, it will be undefined

interface Person {
    firstname ? : string; // the "?" makes the property optional, 
    lastname ? : string; //  so you can start with an empty object
}
export class ContactComponent {

    person: Person = {}; // initializing with an empty object

    constructor(private PeopleService: PeopleService) {}

    ngOnInit() {

        this.PeopleService.fetchData().subscribe(
            data => {
                this.person.firstname = data.results[0].name.first;
                console.log(this.person.firstname);
            });
    }
}

Create a class and inherit in component. For Example see the below code

       export class Person{

            firstname: string;
            lastname: string;
        }
        export class ContactComponent  {
        person=Person[];
        constructor(private PeopleService: PeopleService){
        }

        ngOnInit(){

        this.PeopleService.fetchData().subscribe(
            data=> {
                this.person.firstname=data.results[0].name.first;
                console.log(this.person.firstname);
            });
          }
        }

First declare an object of person type on onInit().

person p=new person();