You should include a hidden CSRF (cross site request forgery) token field in the form so that the CSRF protection middleware can validate the request.
Laravel automatically generates a CSRF "token" for each active user session managed by the application. This token is used to verify that the authenticated user is the one actually making the requests to the application.
So when doing ajax requests, you'll need to pass the csrf token via data parameter.
Here's the sample code.
var request = $.ajax({
url : "http://localhost/some/action",
method:"post",
data : {"_token":"{{ csrf_token() }}"} //pass the CSRF_TOKEN()
});