How do i filter an array inside of a array of objects?

you can use indexOf function

var foo = initialState.filter(function(elm){
  return elm.tags.indexOf("camping")>=0
});

Like that?

const filter = 'nature';
const filteredResult = initialState.filter((item) => {
    return (item.tags.indexOf(filter) >= 0);
});

You can create a Set of selected tags, and use Array#some to check if at least one of the tags in the Set exists in the objects' tags list:

const initialState = [
     {id:1 ,name: 'Product A', image: 'pic-001.jpg', tags: ['nature', 'camping', 'outdoor']},
     {id:2 ,name: 'Product B', image: 'pic-002.jpg', tags: ['winter', 'hiking', 'camping', 'snow']},
     {id:3 ,name: 'Product C', image: 'pic-003.jpg', tags: ['vacation', 'family', 'kids', 'river', 'lake', 'fishing']}
];

const filterByTags = ['nature', 'family'];

const filterByTagSet = new Set(filterByTags);

const result = initialState.filter((o) => 
  o.tags.some((tag) => filterByTagSet.has(tag))
);

console.log(result);

Firstly to get all the tags and filter the duplicate tags from initialState. Save the array into uniqueTags.

Then compare the uniqueTags with initialState names, to create another array productTags with objects and its properties tags and products.

const initialState = [
     {id:1 ,name: 'Product A', image: 'pic-001.jpg', tags: ['nature', 'camping', 'winter', 'outdoor']},
     {id:2 ,name: 'Product B', image: 'pic-002.jpg', tags: ['winter', 'hiking', 'family', 'camping', 'snow']},
     {id:3 ,name: 'Product C', image: 'pic-003.jpg', tags: ['vacation', 'family', 'kids', 'river', 'lake', 'fishing']}
   ]

   let allTags = [];
   initialState.map((t)=>t.tags).forEach((a)=>a.forEach((b)=>allTags.push(b)))
   let uniqueTags = allTags.filter((a,i,arr)=>arr.indexOf(a,i+1)===-1)


   productTags = [];
   uniqueTags.forEach((u)=>{
       initialState.forEach((t)=>{
           if(t.tags.includes(u))
            productTags.push({'tag': u, 'product':t.name});
       })
   })


   console.log(JSON.stringify(productTags));

   /*
   [
    { "tag": "nature", "product": "Product A" }, 
    { "tag": "outdoor", "product": "Product A" }, 
    { "tag": "winter", "product": "Product A" }, 
    { "tag": "winter", "product": "Product B" }, 
    { "tag": "hiking", "product": "Product B" }, 
    { "tag": "camping", "product": "Product A" }, 
    { "tag": "camping", "product": "Product B" }, 
    { "tag": "snow", "product": "Product B" }, 
    { "tag": "vacation", "product": "Product C" }, 
    { "tag": "family", "product": "Product B" }, 
    { "tag": "family", "product": "Product C" }, 
    { "tag": "kids", "product": "Product C" }, 
    { "tag": "river", "product": "Product C" }, 
    { "tag": "lake", "product": "Product C" }, 
    { "tag": "fishing", "product": "Product C" }
] */

(Edited later) Correction:

To form the correct object, I have changed the code to:

const initialState = [
     {id:1 ,name: 'Product A', image: 'pic-001.jpg', tags: ['nature', 'camping', 'winter', 'outdoor']},
     {id:2 ,name: 'Product B', image: 'pic-002.jpg', tags: ['winter', 'hiking', 'family', 'camping', 'snow']},
     {id:3 ,name: 'Product C', image: 'pic-003.jpg', tags: ['vacation', 'family', 'kids', 'river', 'lake', 'fishing']}
   ]

   let allTags = [];
   initialState.map((t)=>t.tags).forEach((a)=>a.forEach((b)=>allTags.push(b)))
   let uniqueTags = allTags.filter((a,i,arr)=>!arr.includes(a,i+1))


   productTags = [];
   uniqueTags.forEach((u)=>{
       let productName = [];
       initialState.forEach((t)=>{
           if(t.tags.includes(u))
            productName.push(t.name);
       })
       productTags.push({tag:u, products:productName}); 
   })

   console.log(JSON.stringify(productTags));

   /*
    productTags = [
        {"tag":"nature","products":["Product A"]},
        {"tag":"outdoor","products":["Product A"]},
        {"tag":"winter","products":["Product A","Product B"]},
        {"tag":"hiking","products":["Product B"]},
        {"tag":"camping","products":["Product A","Product B"]},
        {"tag":"snow","products":["Product B"]},
        {"tag":"vacation","products":["Product C"]},
        {"tag":"family","products":["Product B","Product C"]},
        {"tag":"kids","products":["Product C"]},
        {"tag":"river","products":["Product C"]},
        {"tag":"lake","products":["Product C"]},
        {"tag":"fishing","products":["Product C"]}
    ] 
    */