You're in luck, the API for this is very straight forward (and well described by MDN: https://developer.mozilla.org/en-US/docs/Web/API/Window/localStorage).

Just store your value in the local storage with a key:

localStorage.setItem('your-key', 'your-value')

And afterwards get the value like this:

localStorage.getItem('your-key')

Beware of JSON encoding, though.

/e your example could be restructured like this, in its simples form:

function translate_en(){
localStorage.setItem('lang', 'en')
document.getElementById("intro").innerHTML = `Here at Lion Kuts we are a cat only establishment that offers a full range of services from complete grooming, bathing to boarding. You and your pet will be thrilled to know that only professional, natural and biodegradeable products are used, any sensitivities or allergies will not be a problem.`;
}
   
function translate_fr(){
localStorage.setItem('lang', 'fr')
document.getElementById("intro").innerHTML = `Chez Coupe Lion, nous ne sommes qu'un chat établissement offrant une gamme complète de services du toilettage complet, de la baignade à l'embarquement.Vous et votre animal sera ravi de savoir que seul un professionnel, des produits naturels et biodégradables sont utilisés, tout les sensibilités ou les allergies ne seront pas un problème.`;
}

// The moment this script executes, get the previous language selection, if any.
// Use it to repeat the users selection by automatically calling the right function
// depending on the language the user selected earlier.
const selectedLanguage = localStorage.getItem('lang')
if (selectedLanguage === 'en') {
    translate_en()
} else if (selectedLanguage === 'fr') {
    translate_fr()
}

I would not do it the same way but if you want to keep your structure you can do something like this:

function translate_fr() {
    localStorage.setItem('language', 'FR')
    document.getElementById("intro").innerHTML = `Chez Coupe Lion, nous ne sommes qu'un chat établissement offrant une gamme complète de services du toilettage complet, de la baignade à l'embarquement.Vous et votre animal sera ravi de savoir que seul un professionnel, des produits naturels et biodégradables sont utilisés, tout les sensibilités ou les allergies ne seront pas un problème.`;
} 

function translate_en() {
    localStorage.setItem('language', 'EN')
    document.getElementById("intro").innerHTML = `Here at Lion Kuts we are a cat only establishment that offers a full range of services from complete grooming, bathing to boarding. You and your pet will be thrilled to know that only professional, natural and biodegradeable products are used, any sensitivities or allergies will not be a problem.`;
}

const language = localStorage.getItem('language')
if(language === 'FR') {
    translate_fr()
}

I more answer the part about the translation of the content, as you usually would do this by using a seperate content object. In this object you store all contents and with a js framework or vanilla js, you inject it in a specific paragraph. It is a more scalable approach as you can easily add de or any other language to the object as well, without changing the logic. A small example on how this would work:

const content = {
  en: {
    welcome: 'hi welcome on our website',
    body: 'a beautiful sentence'
  },
  fr: {
    welcome: 'bonjour, bienvenue sur notre site',
    body: 'une belle phrase'
  },
};


let currentLanguage = 'en' // or localStorage.getItem('locale')
const updateContent = (elem, msg) => elem.innerHTML = msg;
const handleLocalization = (locale) => {
  currentLanguage = locale;
  
  // and update localStorage
  // localStorage.setItem('locale', locale);
  updateContent(document.querySelector('#my-welcome-message'), content[currentLanguage].welcome);
    updateContent(document.querySelector('#my-body-message'), content[currentLanguage].body);
}

document.querySelector('#toggle-language').addEventListener('click', () => {
  handleLocalization(currentLanguage === 'en' ? 'fr' : 'en');
})

updateContent(document.querySelector('#my-welcome-message'), content[currentLanguage].welcome);

updateContent(document.querySelector('#my-body-message'), content[currentLanguage].body);

<p id="my-welcome-message"></p>
<p id="my-body-message"></p>

<button id="toggle-language">toggle language</button>