The following code just declares a string variable that contains a MySQL query:

$sql = "INSERT INTO users (username, password, email)
    VALUES ('".$_POST["username"]."','".$_POST["password"]."','".$_POST["email"]."')";

It does not execute the query. In order to do that you need to use some functions but let me explain something else first.

NEVER TRUST USER INPUT: You should never append user input (such as form input from $_GET or $_POST) directly to your query. Someone can carefully manipulate the input in such a way so that it can cause great damage to your database. That's called SQL Injection. You can read more about it here

To protect your script from such an attack you must use Prepared Statements. More on prepared statements here

Include prepared statements to your code like this:

$sql = "INSERT INTO users (username, password, email)
    VALUES (?,?,?)";

Notice how the ? are used as placeholders for the values. Next you should prepare the statement using mysqli_prepare:

$stmt = $mysqli->prepare($sql);

Then start binding the input variables to the prepared statement:

$stmt->bind_param("sss", $_POST['username'], $_POST['email'], $_POST['password']);

And finally execute the prepared statements. (This is where the actual insertion takes place)

$stmt->execute();

NOTE Although not part of the question, I strongly advice you to never store passwords in clear text. Instead you should use password_hash to store a hash of the password

Make sure all post values are getting correctly. You should make a condition check before inserting the data, For ex:

$id = isset($_POST['id']) ? $_POST['id'] : '';
$name = isset($_POST['name']) ? $_POST['name'] : '';
$year = isset($_POST['year']) ? $_POST['year'] : '';

if($id && $name && $year){
$sql = "INSERT INTO cars (id, name, year)
VALUES ($id, '$name', '$year')";
}else{
  return "required fields are missing";
}

NB: Please post your html if possible.