As of PostgreSQL 9.4, you can use the ? operator:

select info->>'name' from rabbits where (info->'food')::jsonb ? 'carrots';

You can even index the ? query on the "food" key if you switch to the jsonb type instead:

alter table rabbits alter info type jsonb using info::jsonb;
create index on rabbits using gin ((info->'food'));
select info->>'name' from rabbits where info->'food' ? 'carrots';

Of course, you probably don't have time for that as a full-time rabbit keeper.

Update: Here's a demonstration of the performance improvements on a table of 1,000,000 rabbits where each rabbit likes two foods and 10% of them like carrots:

d=# -- Postgres 9.3 solution
d=# explain analyze select info->>'name' from rabbits where exists (
d(# select 1 from json_array_elements(info->'food') as food
d(#   where food::text = '"carrots"'
d(# );
 Execution time: 3084.927 ms

d=# -- Postgres 9.4+ solution
d=# explain analyze select info->'name' from rabbits where (info->'food')::jsonb ? 'carrots';
 Execution time: 1255.501 ms

d=# alter table rabbits alter info type jsonb using info::jsonb;
d=# explain analyze select info->'name' from rabbits where info->'food' ? 'carrots';
 Execution time: 465.919 ms

d=# create index on rabbits using gin ((info->'food'));
d=# explain analyze select info->'name' from rabbits where info->'food' ? 'carrots';
 Execution time: 256.478 ms

Try the following:

var categories = [
    {catValue:1, catName: 'Arts, crafts, and collectibles'},
    {catValue:2, catName: 'Baby'},
    {catValue:3, catName: 'Beauty and fragrances'},
    {catValue:4, catName: 'Books and magazines'},
    {catValue:5, catName: 'Business to business'},
    {catValue:6, catName: 'Clothing, accessories, and shoes'},
    {catValue:7, catName: 'Antiques'},
    {catValue:8, catName: 'Art and craft supplies'},
    {catValue:9, catName: 'Art dealers and galleries'},
    {catValue:10, catName: 'Camera and photographic supplies'},
    {catValue:11, catName: 'Digital art'},
    {catValue:12, catName: 'Memorabilia'}
];

var categoriesJson = JSON.stringify(categories);
var mainCat = ['Arts, crafts, and collectibles', 'Baby' , 'Antiques']

$.each(JSON.parse(categoriesJson) , function (key, value) {
  if(mainCat.indexOf(value.catName) > -1){
   console.log('Exists: ' +value.catName)
 }
 else{
   console.log('Does not exists: ' +value.catName)
 }
});

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>

On MySQL 5.7.8+ you can perform a JSON_CONTAINS for each separate value:

SELECT * 
FROM   Schedule 
WHERE  (   JSON_CONTAINS(Days, '1')
        OR JSON_CONTAINS(Days, '2')
        OR JSON_CONTAINS(Days, '6')
       )

When the values to be searched for are stored in a PHP variable -- as in your question -- then you could build the above SQL like this:

$DaysVar = $_GET['Days'];
$condition = implode(" OR ", array_map(function($day) {
    return "JSON_CONTAINS(Days, '".intval($day)."')";
}, $DaysVar));
$sql = mysqli_query($db, "
    SELECT ScheduleID, 
           Days
    FROM   Schedule
    WHERE  ($condition)
");

To avoid SQL injection, you should better prepare your statement creating a string like with substr(str_repeat(" OR JSON_CONTAINS(Days, ?)", count($DaysVar)), 3) and then call mysqli_prepare on it, then bind, ...etc.

MySQL 8.*

Since MySQL 8 you can use JSON_OVERLAPS:

SELECT * 
FROM   Schedule
WHERE  JSON_OVERLAPS(Days, '$DaysVar')

Again, you would better prepare the statement (with a ? placeholder), then bind $DaysVar to it, and finally execute it.