Once you get the response from your API you should then store any non sensitive information that you may need in your front-end. If you are using something like JWT you can store the token in the localstorage and use jwt-decode library for reading it.

    fetch(this.email.value, this.password.value)
    .then(res => {
        localStorage.setItem('id_token', res.token) // Store token 
    })
    .catch(err => {
        console.log(err);
    });

It also pretty common in react to have utility or helper file that handles your authorization

/utils/AuthUtility

    class AuthService{
        login(email, password) {
            return this.fetch('/login', {
                method: 'POST',
                headers: { 'Content-Type': 'application/json' },
                body: JSON.stringify({
                    email, 
                    password
                })
            })
              .then(res => {
                if(res.type == 'success'){
                    this.setToken(res.token) // Setting the token in localStorage
                    return Promise.resolve(res); 
                } else {
                    return Promise.reject(res)
                }
            })
        }


  // Other available methods
       setToken(idToken) {
        // Saves user token to localStorage
        localStorage.setItem('id_token', idToken)
       }

      getProfile() {
        // Using jwt-decode npm package to decode the token
        return decode(localStorage.getItem('id_token'); // assuming you have jwt token then use jwt-decode library
      }
    }

then in your login component //components/login.js

import AuthUtility from './utils/AuthUtility';

login = (e) => {


    this.Auth.login(this.email.value, this.password.value)
    .then(res => {
        this.props.history.push('/protectedRoute');
    })
    .catch(err => {
        console.log(error);
    });
}